RAPHAEL DIAS DA SILVA
Translating one-way quantum computation to
the circuit model: methods and applications
Niterói
September 2013
RAPHAEL DIAS DA SILVA
Translating one-way quantum computation to
the circuit model: methods and applications
Tese apresentada ao Curso de PósGraduação em Física da Universidade
Federal Fluminense, como requisito parcial
para obtenção do Título de Doutor em
Física.
Orientador:
Prof. Dr. Ernesto Fagundes Galvão
Universidade Federal Fluminense
Instituto de Física
Niterói
September 2013
S586t
Silva, Raphael Dias da.
Translating one-way quantum computation to the circuit
model: methods and applications / Raphael Dias da Silva ;
orientador: Ernesto F. Galvão. –- Niterói, 2013.
166 f. : il.
Tese (Doutorado) – Universidade Federal Fluminense,
Instituto de Física,2013.
Bibliografia: f. 159-166.
1.COMPUTAÇÃO QUÂNTICA. 2.CIRCUITO QUÂNTICO. I.Galvão,
Ernesto Fagundes, Orientador. II.Universidade Federal
Fluminense. Instituto de Física, Instituição responsável.
III.Título.
CDD 530.120285
À minha mãe e à minha avó, os pilares da minha vida.
"All that is sacred comes from youth
Dedication, naive and true."
Not For You, Pearl Jam.
Agradecimentos
Eu gostaria de agradecer ao meu orientador Ernesto F. Galvão por dedicar tanto
do seu tempo à minha orientação desde a iniciação científica. Agradeço pelas inúmeras
discussões que tivemos e por suas tentativas de me explicar conceitos físicos com o máximo
de clareza possível. Sua atenção aos detalhes e zelo pela precisão são responsáveis pela
qualidade que esta tese possui. Todas as imperfeições, no entanto, são por minha conta.
Ernesto, obrigado pela sua paciência e dedicação por todos esses anos!
Gostaria também de agradecer à Elham Kashefi, que sempre se mostrou incrivelmente
entusiasmada com minha pesquisa desde o início do meu doutorado. Muito foi aprendido
quando estive sob sua tutela durante meu estágio de doutorado em Edimburgo, Escócia.
A convivência com ela foi muito importante para a minha formação e na decisão de seguir
a carreira acadêmica.
Durante o meu estágio de doutorado no exterior eu pude conhecer várias pessoas
especiais. Agradeço aos meus amigos Einar Pius e Vedran Dunjko por terem me recebido
tão bem em Edimburgo. Obrigado por todas as discussões que tivemos, pelas conversas
sobre a vida e pelas cervejas. Eu aprendi muito com a convivência com vocês dois!
Várias outras pessoas contribuíram para que minha estadia na nublada Edimburgo
fosse mais prazerosa. Dentre elas eu gostaria de agradecer especialmente ao Leonardo
Disilvestro, Tomoyuke Morimae, Riinu Ots, Vincent Danos e à Stefania Bonà.
Gostaria também de agradecer ao Damian Markham pelo convite para apresentar meu
trabalho em Paris e por me receber tão bem durante o período em que visitei seu grupo
de pesquisa. Gostaria também de agradecer ao Simon Perdrix e ao Mehdi Mhalla pelo
interesse em meu trabalho e pelo convite de visitar seu grupo de pesquisa em Grenoble,
França. As discussões que tive com vocês foram muito esclarecedoras e agradáveis.
Gostaria também de agradecer aos meus amigos do IF-UFF que em muito contribuíram para a minha formação como físico e como pessoa. Uma lista incompleta é
formada por Igor Diniz, Carol Vannier, Cadu, Daniel Brod, Dayanne Amaral, Wanessa
Sarzêdas, Alexandre Lima, Thiago Caramês e Cleidson Castro, aos quais agradeço fortemente. Vários professores do IF-UFF tiveram um papel importantíssimo na minha for-
mação acadêmica e também como ser humano, dentre eles gostaria de agradecer especialmente ao Marco Moriconi, Daniel Jonathan, Antônio Zelaquett, Nivaldo Lemos e Antonio
Costa.
Agradeço também aos meus amigos de mais de uma década Bruno Nazar, Karlos
Dyego e Karla Louise. O incentivo e a amizade de vocês têm sido determinante na minha
formação como ser humano.
Gostaria especialmente de agradecer à minha mãe e à minha avó pelo amor incondicional e pela crença que sempre depositaram em mim. Muito obrigado por me darem a
oportunidade de realizar meus sonhos e por nunca permitir que as dificuldades de nossa
família chegassem até mim. Absolutamente tudo que sou e algum dia serei eu devo a
vocês duas. Eu amo vocês!
Por fim agradeço à minha namorada Helena, pelo apoio incondicional que sempre me
deu. Helena, meu amor, obrigado por acreditar em mim nos momentos em que nem eu
acreditava. Sem você ao meu lado tudo teria sido muito mais difícil. Te amo demais!
Acknowledgements
I would like to thank my advisor Ernesto F. Galvão for all the time he has dedicated
to mentoring me, ever since I was an undergrad. I thank him for the numerous discussions, and for his efforts in transmitting physical concepts in a crystal clear manner. His
attention to detail and precision were essential for the quality of this work. All its imperfections, however, are my own responsibility. Ernesto, thank you for your patience and
dedication through all these years!
I would also like to thank Elham Kashefi, who has shown such great enthusiasm in
my research since the beginning of my PhD. I learned a great deal under her mentoring
during my stay in Edinburgh, Scotland. Working with her greatly contributed both to
my academic experience and my decision to pursue this career.
During my stay abroad I was also fortunate to meet a number of special people. I
thank my friends Einar Pius and Vedran Dunjko for receiving me so warmly in Edinburgh.
Thank you for the discussions we had, for the great talks, and for the beers. I learned a
great deal from you guys!
Many others contributed to brighten my stay in rainy Edinburgh. Among them, I
would like to particularly thank Leonardo Disilvestro, Tomoyuke Morimae, Riinu Ots,
Vincent Danos, and Stefania Bonà.
I would also like to thank Damian Markham for inviting me to present my research
in Paris, and for receiving me so warmly during my visit to his research group. I would
also like to thank Simon Perdrix and Mehdi Mhalla for the interest shown in my work,
and for inviting me to visit their research group in Grenoble, France. The talks we had
were both pleasant and enlightening.
I also thank my friends from the Physics Institute at UFF, who greatly contributed
for both my academic and personal growth. An (incomplete) list is Igor Diniz, Carol Vannier, Cadu, Daniel Brod, Dayanne Amaral, Wanessa Sarzêdas, Alexandre Lima, Thiago
Caramês, and Cleidson Castro, all to whom I’m deeply grateful. I am also grateful to
the many teachers who played a major role in my development as a scientist and human
being, particularly Marco Moriconi, Daniel Jonathan, Antônio Zelaquett, Nivaldo Lemos,
and Antonio Costa.
I also thank my longtime friends Bruno Nazar, Karlos Dyego and Karla Louise. Your
friendship and support have immensely contributed to the person I am today.
I would especially like to thank my mother and my grandmother for their unconditional love, and for always believing in me. Thank you for providing me with the
opportunity to follow my dreams, and for never letting our family’s hardships reach me.
Absolutely everything that I am and will ever be I owe it to you. I love you!
Finally, I would like to thank my girlfriend Helena for the ever-present unconditional
support. Helena, my love, thank you for believing in me even when I didn’t believe in
myself. Everything would have been unbearably harder without you by my side. I love
you dearly!
Abstract
In this thesis I study the one-way quantum computation (1WQC) model and some
applications of the different ways of translating 1WQC algorithms into the circuit model.
In a series of recent results, different sets of conditions for implementing a computation
deterministically in the one-way model have been proposed, each of them with their own
properties. Some of those sets of conditions - generically known as flow conditions - try
to explore the distinct parallel power of the 1WQC model, by increasing the number of
operations that can be performed simultaneously. Here I contribute to this line of research
by defining a new type of flow, which I call the signal-shifted flow (SSF), which has an
interesting parallel structure that equals that of a depth-optimal flow.
I also introduce a new framework for translating 1WQC algorithms into the circuit
model. This translation preserves not only the computation performed but also some
features of the 1WQC algorithm design. Within this framework I give two algorithms,
each implementing a different translation procedure: the first gives compact (in space use)
circuits for Regular Flow one-way computations, and the second does the same for SSF
one-way computations. As an application of the SSF translation procedure, I combine it
with other translation and optimization techniques to give an automated quantum circuit
optimization procedure. This procedure is based on back-and-forth translation between
the 1WQC and the circuit model, using 1WQC techniques to time-optimize computations
in the circuit model.
In the second part of this thesis, I use 1WQC tools to analyze quantum circuits
interacting with closed timelike curves (CTCs). I do so by translating to the 1WQC
model CTC-assisted circuits, and then showing that in some cases they can be shown
to be equivalent to time-respecting circuits. The predictions obtained in those cases are
exactly those of the quantum CTC model based on post-selected teleportation, proposed
by Bennett, Schumacher and Svetlichny (BSS). This enabled us to show that the BSS
model for quantum CTCs makes predictions which disagree with those of the highly
influential CTC model proposed by David Deutsch.
Resumo
Nesta tese eu estudo o modelo de computação quântica baseada em medições (CQBM)
e algumas aplicações das diferentes maneiras de traduzir algoritmos de CQBM para o modelo de circuitos. Em uma série de resultados recentes, vários conjuntos de condições para
implementar uma computação deterministicamente no modelo de CQBM têm sido propostas, cada um deles com diferentes propriedades. Alguns desses conjuntos de condições
- genericamente conhecidos como condições de fluxo (flow ) - tentam explorar o poder
de paralelização do modelo de CQBM, aumentando o número de operações que podem
ser realizadas simultaneamente. Aqui eu contribuo para essa linha de pesquisa definindo
um novo tipo de fluxo, chamado fluxo de sinal deslocado (FSD), que tem uma estrutura
paralela interessante que se iguala ao de um fluxo ótimo, do ponto de vista temporal.
Eu também introduzo um novo sistema para traduzir algoritmos de CQBM para o
modelo de circuitos. Esta tradução preserva não só a computação, mas também outras
características de algoritmos em CQBM. Usando esse sistema eu desenvolvo dois algoritmos, cada um capaz de executar um procedimento de tradução diferente: o primeiro
obtém circuitos compactos a partir de computações com fluxo regular, e o segundo faz o
mesmo para computações com FSD. Como uma aplicação do procedimento de tradução de
computações com FSD, eu combino esse procedimento com outras técnicas de tradução
e otimização para desenvolver um procedimento automático de otimização de circuitos
quânticos. Esse procedimento é baseado em traduções nos dois sentidos entre os modelos
de CQBM e de circuitos, usando técnicas de CQBM para otimizar circuitos quânticos
Na segunda parte desta tese, eu uso ferramentas do modelo de CQBM para analisar
circuitos quânticos interagindo com curvas temporais fechadas (CTFs). Essa análise é
feita traduzindo circuitos interagindo com CTFs para o modelo de CQBM e em seguida
mostrando que, em alguns casos, esses circuitos podem ser transcritos como circuitos
sem CTFs que realizam a mesma computação. As predições obtidas nesses casos são
exatamente as mesmas daquelas obtidas usando o modelo para estudar CTFs proposto
por Bennett, Schumacher e Svetlichny (BSS). Isso nos permitiu mostrar que o modelo BSS
para CTFs faz predições que não concordam com aquelas dadas pelo influente modelo de
CFTs proposto por David Deutsch.
List of Figures
1
The application of the unitary evolutions Bell" (a) or Bell# (b) followed
by the measurement of both qubits in the computational basis is equivalent to the measurement of those qubits in the Bell basis. The circuits
in (a) and (b) will be used later as part of the teleportation protocol. .
p. 12
2
The teleportation protocol (Section 2.1.2). . . . . . . . . . . . . . . . .
p. 13
3
The double teleporter: two teleportation protocol circuits (Fig. 2) combined to teleport two spatially separated states |ai and |bi to the same
place. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
A double teleporter with a CN OT gate being applied to the teleported
states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
and |bi using only measurements and classically controlled operations. .
p. 18
Universal graph states: (a) Square Lattice, (b) Triangular Lattice, (c)
Hexagonal Lattice, (d) Kagome Lattice. . . . . . . . . . . . . . . . . . .
7
p. 17
Adapted version of the double teleporter: using the four-qubit state
| iabcd , this circuit applies the CN OT gate to the unknown states |ai
6
p. 16
p. 22
Graphs implementing (a) Arbitrary single-qubit unitary, (b) CN OT gate,
(c) CZ gate, (d) two arbitrary single-qubit rotations followed by a CZ
gate and then other two single-qubit rotations. . . . . . . . . . . . . . .
8
(a) composed J gate teleportation protocol and (b) 1WQC protocol using
adaptive measurements to implement the same unitary as in (a). . . . .
9
p. 30
p. 32
The simulation of a quantum circuit in a cluster state. The black dots
represent qubits measured in the Z basis (and hence deleted from the
cluster) and the circles with arrows represent qubits measured in the B(✓)
basis. Different arrow directions indicate different angles of measurement.
Each set of qubits boxed with a dashed line represents the simulation of
a given circuit gate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 35
10
Examples of graphs satisfying different flow conditions. In Fig. (a) a
graph with regflow is depicted, in (b) a graph with gflow but no regflow
and in (c) a graph with three different gflows: a regflow, a generalized
flow and an optimal gflow (see Section 3.3.6 for more information about
this example). Each vertex represents a qubit, where the black vertices
represent measured qubits and the white ones are unmeasured qubits.
The input qubits are represented by boxed vertices and the arrows denote
the dependency structure, pointing from vertex i to its correcting set.
The dashed line with an arrow represent the case where the vertex from
the correcting set is a non-neighbouring vertex (vertices not linked by an
edge). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
Generic structure of extended circuits obtained from graphs with flow or
gflow. See main text for information about the division into time slices.
12
p. 57
p. 64
Extended circuit for a simple one-way quantum computation protocol.
This J-gate identity will be used repeatedly to simplify generic extended
circuits in Chapters 5 and 6. Note that the J gate angles in (a) and (b)
differ by a minus sign. . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
p. 65
Two isomorphic graphs: (a) generically arranged graph and (b) same
graph re-arranged to obey, from left to right, the partial ordering induced
by regular flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
Quantum circuit obtained from the graph in Fig. 13-b using the Star
Pattern translation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
p. 67
p. 67
Fig. (a) shows a subgraph that works as a building block for the Star
Pattern translation. Decomposing a general graph into these subgraphs,
we can translate each one to the corresponding circuit (Fig. b) and
compose them to obtain the complete translated quantum circuit. . . .
16
p. 68
Using the method in Def. 16 to generate an open graph state able to
simulate the circuit on the top of the figure. . . . . . . . . . . . . . . .
p. 70
17
Graphical representation of the star pattern translation being applied to
the gflow pattern associated to the graph in (a). Figures (b) and (c) show
the star pattern decomposition for the graph in (a). These subgraphs
can be directly translated to the circuit model using the correspondence
shown in Fig. 15. As a result we have the circuits in (d) and (e). Finally,
composing these subcircuits we obtain the quantum circuit in (f). We
can rewrite this circuit, preserving its meaning, to the circuit in (g) by
adding a SW AP gate acting on the (newly added) time-traveling qubit
[the round wire at the bottom of Fig. (g)] and the left-most wire segment
where the anachornical CZ in Fig. (f) were acting upon. By doing so,
the quantum state just before the SW AP gate is able to go to the future,
be acted upon by the CZ gate and then go back to the past and re-enter
the time respecting part of the circuit using the same SW AP gate. The
circuit in (g) has no anachronical CZ like the circuit in (f) but instead
there is a quantum wire interacting with a closed wire (which represents
a qubit traveling in time). . . . . . . . . . . . . . . . . . . . . . . . . .
18
The J-gate identity. This is the same identity as the one shown in Sec.
4.1, and it is reproduced here for convenience of the reader. . . . . . . .
19
p. 72
p. 77
Extended circuit representation of a measurement and the corrections
it requires on other qubits for deterministic computation in the 1WQC
model. Note that each correction arises from a particular stabilizer in
the correcting set of i, raised to the power si , where si is the outcome of
measurement i. This sub-circuit shows only the gates corresponding to
the correcting set of qubit i. . . . . . . . . . . . . . . . . . . . . . . . .
20
p. 79
Five circuit identities. The circuit identity in Figure (b) is true only if
qubit k is initially in the |+i state. The circuit identity in Figure (d)
[Figure (e)] is obtained by multiplying CZik (CXik ) in both sides of the
identity in Figure (a) [Figure (c)]. . . . . . . . . . . . . . . . . . . . . .
21
p. 80
Rewrite Procedure 1. In Fig. 21-a we show the undesired CZik in time
slice Ci with the corresponding, initial entangling-round CZjk in time
slice E1 . In Fig. 21-b the CZjk gate was moved from E1 to Ci where
the circuit identity in Fig. 20-a can be applied, resulting in the circuit
22
depicted in Fig. 21-c. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 82
Stepwise influencing path }i (j). See Def. 21. . . . . . . . . . . . . . . .
p. 96
23
Extending a stepwise influencing path ending at vertex v according to
Lemma 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
p. 98
Rewrite Procedure 2. This RP is composed by several applications of
the rewrite rule in Figure 20-d. Although all E gates are drawn in the E
slice, the requirement for this RP to be applied is that those E gates are
placed just before the corresponding CX gates (that is, with no gate in
between). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
Rewrite Procedure 3. If Ls (k) = Ls (i), slices (E, J, C) and (E0 , J0 , C0 )
become the same; The rewrite procedure remains exactly the same. . .
26
p. 99
Rewrite Procedure 4. If Ls (k) = Ls (i), slices (E, J, C) and (E0 , J0 , C0 )
become the same; The rewrite procedure remains exactly the same. . .
27
p. 99
p. 100
Removing even many CZij from a SSF extended circuit. See Lemma 8
for more information. . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 102
28
Using compactification procedures to obtain J-gate parallelization for
circuits with Pauli angles. See example 2 for more information. . . . . .
29
Simplifying an extended circuit originated from the entanglement graph
with gflow of Fig. (a). For a step-by-step explanation, see section 5.4.2.
30
p. 116
Aplication of the procedure described in Def. 16 to the circuit in Figure
32-a. See Step 1 of Sec. 6.1.1 for more information. . . . . . . . . . . .
32
p. 115
Simplifying the extended circuit originated from the graph in Fig. (a).
For a step-by-step explanation, see Sect. 5.4.3. . . . . . . . . . . . . . .
31
p. 113
p. 121
A complete example of global circuit optimization. In each figure the
shaded gates are the new gates obtained through the RPs (see Section
6.1.1 for more information). The angles of each J-gate are arbitrary and
they have been omitted from the figure. . . . . . . . . . . . . . . . . . .
33
p. 125
A summary of the optimization procedure using extended translation and
compactification. Each arrow on the left of the table indicates a different
optimization or translation procedure. The parallelization procedure described in [23] (without Pauli simplification) ends with the “Parallelized
Circuit” whereas our optimization can end with a “Compactified Circuit” or a “Parallelized Compactified Circuit” depending the goal of the
optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 127
34
a) CTC takes qubit back in time to interact with its past self; b) Bennett/Schumacher/Svetlichny (BSS) circuit to simulate this CTC probabilistically, using teleportation and post-selection (see main text). . . .
35
p. 134
Rewritten BSS circuit using preparation of and projections onto |+i
states. The unitary V is decomposed using the universal gate-set consisting of J(✓) and CZ. . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 135
36
BSS circuit simulating a CTC with unitary V = [J( ✓) ⌦ I]CZ. . . . .
p. 135
37
Two equivalent circuits, i.e. implementing the same unitary. In a) we
have a classically-controlled X unitary dependent on the measurement
outcome of |±✓ i basis projection. In b) this has been turned into a
coherent circuit with measurement onto the Z basis. . . . . . . . . . . .
38
p. 137
a) Circuit that includes a CTC with an anachronical CZ gate; it is
equivalent to the two circuits in Fig. 37. b) The same circuit rewritten
in the BSS format. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
p. 137
Circuit that implements the probabilistic BSS simulation of the CTC
circuit in Fig. 38-b. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 138
40
(a) Entanglement graph corresponding to state |Gi in Eq. (7.4). On this
state we can perform the sequence of one-way operations in Eq. (7.3).
The extended translation of Eqs. (7.8), (7.9) and (7.10) are shown in
(c), (e) and (g), respectively. In Figs. (d), (f) and (h) I redraw the CTC
circuits in Figs. (c), (e) and (g), respectively, so as to explicitly show the
CTC as a time-travelling qubit. . . . . . . . . . . . . . . . . . . . . . .
41
Translating the BSS circuit in Fig. 39 to the one-way model (see Sec.
7.4.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
p. 140
p. 144
Stabilizer manipulations that simplify the one-way sequence (7.11); hi +|
denotes a projection onto |0i + i|1i. a) Graph representing initial entanglement structure and operations. Qubit 3 is the input and qubits 1 and
5 are measured in the X basis. b) Effect of local complementation (LC)
on qubit 5; c) LC on qubit 1; d) Pauli Z deletion of qubit 1; 3) LC on
qubit 5; f) Pauli Z deletion of qubit 5. The final pattern represents the
one-way two-qubit implementation of the J( ✓) gate, see Fig. 37. . . .
p. 145
43
An analysis of the stabilizers in eqs. (7.16) and (7.17) indicates that the
graph on the right represents state | i and the one on the left state | i
(see main text). The local complementation unitary C1 S2† S3† changes the
graph on the left into the graph on the right. Local complementations
change the state without changing its entanglement structure or the oneway computations implementable by it. . . . . . . . . . . . . . . . . . . p. 147
44
a) Deutsch’s model for a CTC. b) This is the relationship between Deutsch’s
unitary U and unitary V in the BSS CTC circuit of Fig. 34-b. . . . . .
45
46
p. 148
Three representations for the same CTC circuit. a) Deutsch formulation.
b) BSS formulation. c) Short-hand form of either. . . . . . . . . . . . .
p. 149
Deutsch’s formulation of the extended CTC circuit of Fig. 38-b. . . . .
p. 150
Contents
p. 1
1 Introduction
1.1
General picture and structure of this thesis . . . . . . . . . . . . . . . .
p. 4
1.2
List of publications and corresponding chapters . . . . . . . . . . . . .
p. 8
1.3
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 9
p. 10
2 Quantum computation driven by measurements
2.1
Teleportation-based quantum computation . . . . . . . . . . . . . . . .
p. 11
2.1.1
Bell states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 11
2.1.2
The teleportation protocol . . . . . . . . . . . . . . . . . . . . .
p. 12
2.1.3
Teleportation as a quantum computing primitive
. . . . . . . .
p. 14
2.1.3.1
Teleporting a single-qubit gate . . . . . . . . . . . . .
p. 14
2.1.3.2
The CNOT gate . . . . . . . . . . . . . . . . . . . . .
p. 15
2.2
The rise of measurement-based models . . . . . . . . . . . . . . . . . .
p. 18
2.3
A brief description of the one-way QC model . . . . . . . . . . . . . . .
p. 19
2.4
Resource states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 22
2.4.1
Cluster states and graph states . . . . . . . . . . . . . . . . . .
p. 23
Measuring graph state qubits . . . . . . . . . . . . . . . . . . . . . . .
p. 26
2.5.1
Single-qubit measurements . . . . . . . . . . . . . . . . . . . . .
p. 28
2.5.2
Removing redundant qubits . . . . . . . . . . . . . . . . . . . .
p. 29
2.5.3
Moving the logical state of a qubit in a graph . . . . . . . . . .
p. 29
Simulating the circuit model . . . . . . . . . . . . . . . . . . . . . . . .
p. 30
2.5
2.6
2.6.1
2.7
Universal single-qubit rotations and the need for adaptive measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 31
2.6.2
Two-qubit gates . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 33
2.6.3
Simulating arbitrary quantum circuits
. . . . . . . . . . . . . .
p. 34
2.6.4
Beyond quantum circuit simulations . . . . . . . . . . . . . . . .
p. 36
2.6.5
Simulating quantum circuits with unbounded fan-out . . . . . .
p. 36
Clifford computations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 37
2.7.1
Pauli measurements and the Clifford group . . . . . . . . . . . .
p. 37
2.7.2
The Gottesman-Knill theorem . . . . . . . . . . . . . . . . . . .
p. 38
2.7.3
Quantum-classical tradeoff . . . . . . . . . . . . . . . . . . . . .
p. 39
p. 41
3 Deterministic computation using quantum measurements
3.1
3.2
3.3
The measurement calculus . . . . . . . . . . . . . . . . . . . . . . . . .
p. 43
3.1.1
Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 43
3.1.2
Measurement patterns . . . . . . . . . . . . . . . . . . . . . . .
p. 44
3.1.3
Command identities . . . . . . . . . . . . . . . . . . . . . . . .
p. 47
3.1.4
Standardization . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 48
Example 1: Teleportation . . . . . . . . . . . . . . . . . . . . .
p. 49
Example 2: Single-qubit rotation . . . . . . . . . . . . . . . . .
p. 50
Determinism in 1WQC . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 50
3.2.1
Types of determinism . . . . . . . . . . . . . . . . . . . . . . . .
p. 54
Flow theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 55
3.3.1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 56
3.3.2
Regular flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 56
3.3.3
Generalized flow
. . . . . . . . . . . . . . . . . . . . . . . . . .
p. 58
3.3.4
Maximally-delayed Generalized flow . . . . . . . . . . . . . . . .
p. 59
3.3.5
Constructing a measurement pattern from a flow function
p. 60
. . .
3.3.6
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Translating one-way patterns into the circuit model
p. 60
p. 62
4.1
Extended translation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 63
4.2
Star pattern translation . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 66
4.2.1
From measurement patterns to circuits . . . . . . . . . . . . . .
p. 66
4.2.2
From circuits to measurement patterns . . . . . . . . . . . . . .
p. 69
4.3
4.4
The problem of using star pattern translation for patterns with no regular
flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 71
Alternative approaches to the translation problem . . . . . . . . . . . .
p. 74
5 Compact circuits from one-way quantum computation
5.1
5.2
5.3
p. 75
Compactification procedures and circuit rewriting rules . . . . . . . . .
p. 76
5.1.1
Compactification procedures . . . . . . . . . . . . . . . . . . . .
p. 77
5.1.2
Circuit rewrite rules . . . . . . . . . . . . . . . . . . . . . . . .
p. 78
Compact circuits from graphs with regular flow . . . . . . . . . . . . .
p. 80
5.2.1
Regular flow structural properties . . . . . . . . . . . . . . . . .
p. 81
5.2.2
A compactification algorithm for regular flow extended circuits .
p. 81
5.2.2.1
A rewrite procedure for regular flow . . . . . . . . . .
p. 82
5.2.2.2
The algorithm . . . . . . . . . . . . . . . . . . . . . .
p. 83
5.2.2.3
Comparison with star pattern translation . . . . . . .
p. 85
Compact circuits from graphs with signal-shifted Flow . . . . . . . . .
p. 85
5.3.1
Signal-shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 86
5.3.1.1
Rewrite rules . . . . . . . . . . . . . . . . . . . . . . .
p. 86
Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 87
5.3.1.2
Algorithm . . . . . . . . . . . . . . . . . . . . . . . . .
p. 87
5.3.2
The signal-shifted flow . . . . . . . . . . . . . . . . . . . . . . .
p. 90
5.3.3
SSF structural properties . . . . . . . . . . . . . . . . . . . . . .
p. 95
5.4
5.5
5.3.4
Rewrite procedures for signal-shifted flow . . . . . . . . . . . . .
p. 98
5.3.5
Compactification procedure for signal-shifted flow . . . . . . . .
p. 106
5.3.6
Discussion of results and possible extensions . . . . . . . . . . .
p. 111
Example: Pauli measurements . . . . . . . . . . . . . . . . . . .
p. 111
Compact circuits from graphs with generalized flow . . . . . . . . . . .
p. 113
5.4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 114
5.4.2
First example . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 115
5.4.3
Second example . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 116
5.4.4
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 116
Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 117
6 Optimizing quantum circuits using one-way quantum computation
techniques
p. 118
6.1
Optimizing quantum circuits by back-and-forth translation . . . . . . .
p. 118
6.1.1
Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 120
Step 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 122
Step 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 122
Step 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 123
Step 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 124
Complexity analysis and discussion . . . . . . . . . . . . . . . . . . . .
p. 126
6.2
7 Closed timelike curves in one-way quantum computation
7.1
Review of the literature
7.2
A model for CTCs based on teleportation and post-selection: The BSS
7.3
7.4
. . . . . . . . . . . . . . . . . . . . . . . . . .
p. 130
p. 131
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 133
Closed timelike curves in one-way quantum computation . . . . . . . .
p. 136
7.3.1
Obtaining CTC-assisted circuits from one-way patterns . . . . .
p. 139
Deterministic simulations of CTCs . . . . . . . . . . . . . . . . . . . .
p. 141
7.4.1
7.5
7.6
Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 144
Comparing the BSS model with Deutsch’s model . . . . . . . . . . . .
p. 148
7.5.1
The Deutsch model . . . . . . . . . . . . . . . . . . . . . . . . .
p. 148
7.5.2
Comparison with the BSS model . . . . . . . . . . . . . . . . .
p. 149
7.5.3
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p. 151
One-way model: a toy model for space-time? . . . . . . . . . . . . . . .
p. 152
8 Conclusion and further research directions
p. 155
p. 159
1
1
Introduction
Quantum computers are expected to outperform their classical counterparts. Since
Richard Feynman’s seminal 1982 paper [46], where the idea of a computer based on
quantum physics was first suggested, several different models for quantum information
processing have been proposed. Since there is no rigorous proof that quantum computation is more powerful than classical computation, a lot of work has been done trying to
understand which physical properties give quantum computers their apparent power.
A number of useful quantum algorithms are known, including algorithms for factoring
[105, 110], searching unstructured databases [57] and to simulate quantum systems [13, 74,
47, 16, 72, 69]. However, the lack of understanding about what physical properties enable
a quantum computer to run algorithms that outperform classical ones makes further
development of quantum algorithms a great challenge. A good approach to improve this
understanding is the development and analysis of different models for quantum computing,
in special those relying on different physical properties to drive the computation.
There are several models for quantum computing, such as the quantum circuit model
[38, 86], adiabatic quantum computation [45, 68, 17, 101], and measurement-based quantum computation (MBQC) models such as teleportation-based quantum computation [52],
ancilla-driven quantum computation [104, 11, 67] and the one-way quantum computation
(1WQC) model [95, 21, 97, 100, 20]. Those models were proved to be polynomialy equivalent in terms of depth and spatial resource, but the key elements and requirements for
their physical implementation vary significantly. A particularly interesting class of models is the one that rely on measurements to drive the computation, in special the 1WQC
model. In this model entanglement is a resource provided at the beginning and which is
destroyed by single-qubit measurements, whose correlations enable the computation.
In this thesis I study the one-way model, in particular several issues that arise when
we translate computations from the circuit model to 1WQC model and vice-versa, with
the purpose of getting new insights on how the algorithms work and what key elements are
2
required for the implementation of a given algorithm. I develop a new translation framework able to translate to the circuit model a large class of 1WQC algorithms. The main
difference with respect to other translation procedures is that the method described in this
thesis preserves some features of the 1WQC algorithm as, for example, the dependency
structure between the gates/operations. It is clear that having translation procedures
able to preserve different properties can just enrich the understanding on which resources
are needed to perform a given computation.
I also give two applications that profit from the methods developed for optimizing and
translating one-way computations to the circuit model. The first is a circuit optimization
scheme based on back-and-forth translation between the 1WQC model and the circuit
model. Quantum circuit optimization is a subject of great importance for quantum computing. The enormous technological effort required to perform (even simple) quantum
computing tasks makes the research on circuit optimization a potential shortcut to more
complex quantum computing tasks, bringing quantum algorithms down to the current
technological capabilities.
I contribute to this line of research by introducing an automated circuit optimization
procedure. This procedure is based on the translation of a given quantum circuit into
a one-way quantum computation. These two models utilize remarkably different information processing tools: while the former is based on unitary evolution of an initially
non-entangled set of qubits, the latter needs an initial highly-entangled multi-qubit state,
where the information processing is driven by measurements only. Naturally, since the
two aforementioned models use different information processing tools, each model has its
own optimization techniques.
In the 1WQC model, for instance, most of the optimization techniques are based on the
identification of a more efficient correction structure that is directly linked to the geometry
of the underlying global entanglement structure. Examples of these techniques are signal
shifting [35] and generalised flow [24]. The so-called standardisation procedure [35] can
also reduce the number of computational steps by rearranging the 1WQC operations into
a normal form. Moreover, all Pauli measurements in this model can be performed at
the beginning of the computation [96], which is a surprising difference from the quantum
circuit model.
On the other hand, most optimization techniques for quantum circuits are based on
template identification and substitution. For instance in [49], some circuit identities are
used to modify the teleportation and dense coding protocols, with the purpose of giving a
3
more intuitive understanding of those protocols. Similarly in [103] and [79] a set of circuit
identities for reducing the number of gates in the circuit for size optimization was given.
In contrast to that, in [84] a useful set of techniques for circuit parallelization was provided, where the number of computational steps is reduced by using additional resources.
However, as noted in [103], all the aforementioned circuit optimization techniques are
basically exchanging a sequence of gates for a different one without any consideration of
the structure of the complete circuit being optimised. The translation into 1WQC would
allow us to explore the global structure of a given circuit.
The first such optimization scheme by back and forth translation between the two
models was presented in [23]. However the backward translation into the circuit required
the addition of many ancilla qubits. In this thesis I propose a new automated optimization
scheme able to translate the optimized computation back to the circuit model without
adding any ancillas. We start by translating the circuit to the 1WQC model using a
technique that gives a graph whose geometry reflects the gate arrangement in the associated circuit. We then use a 1WQC optimization technique (known as signal shifting)
able to reduce the depth of the computation1 . It is important to note that, since this
optimization technique relies on global properties of the graph (related to its geometry),
the optimization implemented in the circuit model is related to the global arrangement of
gates in the original circuit. The optimized 1WQC computation is then translated to the
circuit model using a new translation procedure (developed in Chapter 5), resulting in a
circuit with as many wires as the original one but with smaller number of computational
time slices.
The second application is related to a more fundamental topic, namely the existence
and computational power of closed timelike curves. The possibility of time travel has
been studied for decades in the context of general relativity. Several decades after Einstein
developed the general theory of relativity, the first explicit spacetime geometry containing
closed time-like curves (CTCs) - paths in spacetime that would allow a physical system
to interact with its former self - was proposed by Kurt Gödel [50]. After Gödel’s work, a
variety of spacetimes containing CTCs were proposed [18, 51]; in fact, it was later realized
that CTCs are a generic feature of highly curved, rotating spacetimes [18, 109, 71].
Assuming that closed timelike curves exist, a series of results were obtained regarding
their implications for quantum mechanics and quantum computation [39, 12, 28, 15, 92,
93]. In this thesis I describe how the one-way model of measurement-based quantum
1
That is, the same computation can be implemented faster, by increasing the number of operations
that can be performed simultaneously.
4
computation [95] encompasses in a natural way a model for CTCs proposed by Bennett
and Schumacher [1], and more recently (and independently) by Svetlichny [106]. I show
that the one-way model effectively simulates deterministically a class of CTCs in this
model, and characterize this class. A second model for CTCs is Deutsch’s highly influential
study of quantum time-travel [39]. I show that Deutsch’s model leads to predictions
conflicting with those of the one-way model, and identify the reason behind this.
1.1
General picture and structure of this thesis
Here I summarize the main contributions of this thesis and discuss how this thesis is
organized. In the first chapters I review the key ideas of the one-way quantum computation
model, with a special attention to the elements necessary for understanding the original
work presented in later chapters.
• Chapter 2. I start by discussing the seminal work by Gottesman and Chuang
[52] where they show that an adaptation of the teleportation protocol [14] can be
used as a quantum computing primitive. Their scheme, known as gate teleportation or teleportation-based quantum computation, contains the main ingredients of
a measurement-based quantum computation model: preparation of a (sufficiently
large) highly entangled state and measurements on bases which may depend on previously obtained outcomes (adaptive measurements). I proceed with a brief overview
of other measurement-based models that were proposed after [52] and then I present
the one-way quantum computation model [95, 21, 97]. In Sec. 2.4 I give some examples of quantum states that can be used as resource states in a one-way computer,
giving special attention to the widely used graph states. In Sec. 2.5 I show the effect
of performing single-qubit measurements in graph states and in Sec. 2.6 I show how
quantum circuits can be simulated in the one-way model. Finally, in Sec. 2.7 I
discuss the role of Pauli measurements in the one-way model; these measurements
have the interesting property of being independent of any other measurement in a
one-way quantum computation. As a consequence, all Pauli measurements can be
performed at once in the first computational time slice.
One of the most important concepts behind the one way model is the way intrinsically
random quantum measurements can be harnessed as a tool to implement deterministic
computation. Explaining this and some tools to design 1WQC algorithms is the main
goal of the next chapter.
5
• Chapter 3. In Sec. 3.1 I review the so-called measurement calculus [35], which is a
formal language developed to represent and analyze 1WQC algorithms. In Sec. 3.2 I
describe the different types of determinism that a 1WQC algorithm can be required
to have, explaining why I am going to require just one of those types of determinism. Next, in Sec. 3.3, I review several correcting strategies generically known as
flows. A flow is a set of conditions over a graph (representing the initial entanglement structure) that verify whether there exists a partial ordering to perform the
measurements so that a deterministic computation can be implemented. I review
three different types of flow: (1) Regular Flow [33], (2) Generalized Flow [24] and
(3) Maximally-delayed Generalized Flow [81]. Those properties are not mutually
exclusive and therefore a graph can have two or more types of flow. A comparison
between all the aforementioned flows (together with some examples) concludes the
chapter.
The one-way quantum computation model is equivalent to the circuit model. This
means that every problem that is solvable in one model, can also be solved by the other
with at most a polynomial overhead in the number of computational steps. However, a
computation in the circuit model requires completely different resources than the corresponding computation in the one-way model. Based on this notion, most of the original
work presented in this thesis aims at (i) developing new translation procedures from the
one-way model to the circuit model (Chapter 5) and (ii) translating computations from
one model to another in order to get insights that would be hard to get when analyzing
just one model (Chapters 6 and 7).
• Chapter 4. In this chapter I review two different translation procedures and explore their limitations and advantages. In Sec. 4.1 I review and extend the definition
of extended circuits, which are easily obtainable translations of measurement patterns. As we shall see, this is a straightforward method but it is inefficient in some
important ways. Next, in Sec. 4.2, I describe the so-called star pattern translation
(or simply SP T ). The SP T is a translation framework that gives circuits which
are less demanding (in terms of number of qubits) than the corresponding extended
circuits. The SP T is, however, a very limited translation framework since it works
only for measurement patterns obtained from a regular flow (Sec. 3.3.2). In Sec.
4.2.2, I show how the SP T can be used “backwards” to translate circuits to the
1WQC model, resulting in measurement patterns with regular flow. In Sec. 4.3, I
explain exactly why and in which cases the SP T fails. The insights contained in
6
that section were the starting point for the development of a new translation framework (introduced in Chapter 5 and with an application given in Chapter 6) and the
study of the simulation of closed timelike curves using the 1WQC model (Chapter 7).
• Chapter 5. This chapter is divided as follows. In Section 5.1 I introduce the concept of compact circuits and the method for obtaining them from extended circuits,
which I call compactification procedures. Section 5.2 is reserved for the development of the compactification procedure for regular flow extended circuits. First, in
Sec. 5.2.1, I review the regular flow definition and comment on some important
properties of regular flow extended circuits. The algorithm that implements the
compactification procedure for regular flow is introduced in Sec. 5.2.2; A comparison with the existing method for regular flow translation, known as the Star Pattern
Translation (see Sec. 4.2.1), and an example of the algorithm running are also provided in this section. In the beginning of Section 5.3, I introduce a new type of flow
that I call Signal-shifted Flow (SSF) and explore several of its structural properties.
The SSF is more general than regular flow and its limitations and advantages are
explored in the section. An algorithm to implement the compactification procedure
for SSF extended circuits is given in Sec. 5.3.5. Finally, in Sec. 5.4 I comment on
the difficulty of designing compactification procedures for arbitrary flows and give
a couple of examples of generalized flow extended circuits being transformed into
their compact versions.
In the next two chapters I present two problems whose understanding profits from
the translation between models that I develop. In Chapter 6 I present a quantum circuit optimization procedure. The procedure consists in three parts: (i) translation of
a circuit to the 1WQC model, (ii) application of optimization techniques to the 1WQC
algorithm obtained in (i), and (iii) translation of the optimized 1WQC algorithm back
to the circuit model. The second successful case where the approach of translating algorithms from one model to another brought some new insights, is related to the so-called
closed timelike curves. The analysis of quantum circuits interacting with closed timelike
curves is the subject of Chapter 7.
• Chapter 6. In this chapter I give an application of the SSF compactification procedure introduced and explored in Chapter 5, namely the optimization of quantum
circuits. In Sec. 6.1, I describe how a compactification procedure can be used -
7
together with other techniques - to optimize quantum circuits and give a full example of such procedure. I conclude this chapter with a complexity analysis (Sec.
6.2) of the optimization technique proposed here, comparing it with other known
techniques.
• Chapter 7. I start by giving a brief summary of the recent revival of interest in
closed timelike curves (CTCs) in the quantum information community, highlighting the main contributions over the last years. Then, in Sec. 7.2, I review the
CTC model based on quantum teleportation and postselection proposed by Bennett, Schumacher and Svetlichny [1, 106]. In Sec. 7.3 I discuss how CTCs appear
naturally in the one-way model, and show that they correspond to CTCs in the
Bennett/Schumacher/Svetlichny model. In Sec. 7.4 I characterize a class of CTCassisted quantum circuits that can be rewritten as time-respecting circuits (i.e., with
no CTCs). I give an explicit method for verifying if a CTC-assisted circuit can be
simulated using the 1WQC model. Next, in Sec. 7.5, I review the highly influential
CTC model proposed in 1991 by David Deutsch [39]; I use Deutsch’s consistency
conditions to calculate the predictions for the same CTC-assisted circuits analyzed
in Sec. 7.2 using the BSS model. I show that while the BSS’s model encompasses
the one-way model prediction, Deutsch’s model gives completely different predictions. Finally, in Sec. 7.6 I discuss a recent proposal by Raussendorf et al. [99] for
using 1WQC as a quantum mechanical toy model for space-time.
• Chapter 8. In this final chapter, I give my concluding remarks about the work
presented in this thesis and point out some further research directions and open
problems.
To conclude this section, I would like to make a disclosure. Some of the results
presented in Sec. 5.3, which is based in [42], are due to one of my collaborators in that
work, Einar Pius. More specifically, the algorithm in Sec. 5.3.1.2 and most of the proofs
in Secs. 5.3.2 and 5.3.3 were due to him and presented in our joint work [42]. Since those
results are indispensable for understanding the rest of the results in Sec. 5.3, I have opted
to reproduce (with a few changes and adaptations) part of the text from our joint work
[42].
8
1.2
List of publications and corresponding chapters
I list below the articles (published or submitted) written to communicate the original
work that I present in this thesis. At the end of each item I inform in which chapter(s)
of this thesis one can find the content related to the corresponding article.
1. Raphael Dias da Silva, Ernesto F. Galvão and Elham Kashefi. Closed timelike curves in measurement-based quantum computation Phys. Rev. A
83, 012316 (2011); [Content related to this publication can be found in Chapter 7].
2. Raphael Dias da Silva and Ernesto F. Galvão. Compact quantum circuits from
one-way quantum computation. Phys. Rev. A 88, 012319 (2013); [Content
related to this publication can be found in Chapters 4 and 5].
3. Raphael Dias da Silva, Einar Pius and Elham Kashefi. Global quantum circuit
optimization. Submitted. arXiv:1301.0351 [quant-ph] (2013); [Content related to
this preprint can be found in Chapters 5 and 6].
9
1.3
Notation
In this section I summarize the main notation and symbols that will be used throughout this thesis. I list below the single-qubit gates that I will refer to in this thesis by
its name or symbol (in rare cases I will explicitly show its matrix form). In order of
appereance, the gates are: Hadamard (H), phase (S),
⇡
-gate
8
(T ) and arbitrary phase
gate [P (✓)]:
1
H=p
2
1
1
1
1
!
,
1 0
S=
0 i
!
,
1
T =
0
0 ei⇡/4
!
,
P (✓) =
1
0
0 ei✓
!
.
I will also constantly refer to the so-called Pauli matrices or, in the context of quantum
computing, Pauli gates:
1
Z=
0
0
1
!
,
Y =
0
i
i
0
!
,
X=
0 1
1 0
!
.
The following two-qubit gates will also be constantly used throughout this thesis: ControlledZ (CZ), Controlled-N OT (CX) and the SW AP gate:
0
1 0 0
B
B0 1 0
B
CZ = B
B0 0 1
@
0 0 0
0
1
C
0C
C
C,
0C
A
1
0
1 0 0 0
1
B
C
B0 1 0 0 C
B
C
CX = B
C,
B0 0 0 1 C
@
A
0 0 1 0
0
1 0 0 0
1
B
C
B0 0 1 0 C
B
C
SW AP = B
C.
B0 1 0 0 C
@
A
0 0 0 1
10
2
Quantum computation driven by
measurements
Quantum computation was first conceived as a generalization of classical computing:
most quantum models were built upon key concepts from classical information processing architectures, as it is the case for quantum Turing machines, quantum circuits and
quantum random walks. These models use reversible, unitary evolution to drive the information processing, whereas measurements take place at the end to read out classical information. Due to the irreversible and probabilistic character of quantum measurements, it is
quite surprising that it can be used as main tool in a quantum computing model. However,
in 1999, Gottesman and Chuang [52] described how one could apply arbitrary quantum
gates via measurements, using an adaptation of the quantum teleportation protocol [14].
This approach was further developed by other researchers [87, 73, 63, 64], enabling one
in principle to perform arbitrary computations given a few primitives: preparation of
maximally entangled systems of fixed, small dimension; multi-qubit measurements on arbitrary sets of qubits; and the possibility of adapting the measurement bases depending on
earlier measurement outcomes. These models of computation draw on measurements to
implement the dynamics, and as such are collectively called measurement-based quantum
computation (MBQC).
One of the most prominent MBQC models, the so-called one-way quantum computation (1WQC) model, was developed by Robert Raussendorf and Hans Briegel in 2001
[95]. The one-way model has some remarkably interesting properties: in this model, any
algorithm can be implemented using a generic resource state called the cluster state, upon
which adaptive single-qubit measurements are performed. Moreover, since no entangling
measurement is performed in this model (as is necessary in teleportation-based models),
the contribution of entanglement for the quantum speedup, one of the main fundamental
questions in quantum computing, can be more systematically analyzed by studying which
quantum states are useful resource states.
11
In this chapter I review the main concepts of measurement-based QC models, giving
special attention to the 1WQC model, which is the backbone of this thesis. First, in
Section 2.1, I review the well-known teleportation protocol and explain how it can be
adapted to be used as a quantum computing primitive. This modified teleportation
protocol was the first proposed MBQC model and, as such, it contains the main ideas
behind MBQC models in general. In Section 2.2, I give a brief summary of the different
MBQC models that appeared in the literature soon after the teleportation-based model. In
Section 2.3, a brief overview of the one-way model is given, whereas an in-depth treatment
of the concepts discussed in this section will be given in the following sections. In Section
2.4, I discuss which resource states are useful for the 1WQC model, showing examples of
universal resource states and how to create the widely used graph states. In Section 2.5,
I review projective measurements and show the effect of measuring graph state qubits
in different bases. In Section 2.6, I explain how information is processed in the 1WQC
model and how it can simulate the circuit model. Finally, in Section 2.7, I discuss the
role of the so-called Clifford gates, which surprisingly can be implemented all at once in
the beginning of the computation in the 1WQC model.
2.1
Teleportation-based quantum computation
Many of the important properties of MBQC models are present in the teleportation
protocol [14] itself: the preparation of an entangled state to be used as a resource; the
necessity of communicating and/or storing classical data (measurement outcomes) and
also the local corrections needed to account for the intrinsic random character of measurement. Let us briefly review how the teleportation protocol works and then explain
how it can be used as a primitive for quantum computing.
2.1.1
Bell states
Bell states are maximally-entangled two-qubit states widely used in several quantum
information protocols. Those states were named after John Bell, who explored via the socalled Bell inequalities many interesting properties of such states. They are also known
as EPR-like states, after Einstein, Podolsky and Rosen, which used similar states in a
thought experiment designed to criticize some aspects of quantum theory [44].
Bell states can be constructed in the following way: (1) Prepare the four computational
basis states |00i, |01i, |10i and |11i; (2) apply to each of those states the operator U =
12
H
Bell
Bell
H
(a)
(b)
Figure 1: The application of the unitary evolutions Bell" (a) or Bell# (b) followed by the
measurement of both qubits in the computational basis is equivalent to the measurement
of those qubits in the Bell basis. The circuits in (a) and (b) will be used later as part of
the teleportation protocol.
(H ⌦ I)CN OT , where H is the Hadamard gate and I is the Identity operator. In Fig. 1
I give two circuits able to create Bell states, using a graphical notation that will be used
later in this Chapter. The resulting states are the following:
1
⌘p
2
1
| 01 i ⌘ p
2
1
| 10 i ⌘ p
2
1
| 11 i ⌘ p
2
|
00 i
which can be succinctly represented as |
and v̄ = 1
|00i + |11i
(2.1)
|00i
|11i
(2.2)
|01i + |10i
(2.3)
|01i
(2.4)
vw i
⌘
|10i
p1
2
|0vi + ( 1)w |1v̄i , where v, w 2 {0, 1}
v. Since U is a unitary operator, the four Bell states form a basis for
the Hilbert space of a two-qubit system. A measurement onto this basis is called a Bell
measurement. Note also that any Bell state can be obtained from any other Bell state by
applying Pauli operators:
|
vw i
= Z w X v ⌦ I|
00 i
= I ⌦ ZwX v|
00 i
(2.5)
up to a global phase. In other words, if two parties share a Bell state, each party has the
ability of transforming it into another Bell state by applying local Pauli operators.
2.1.2
The teleportation protocol
The teleportation protocol [14] consists basically of three steps: (1) An EPR pair is
shared between Alice and Bob; (2) Alice, who wants to send an unknown state | i to Bob,
13
s1
| i
|
Bell
s2
00 i
X
Z
| i
Figure 2: The teleportation protocol (Section 2.1.2).
performs a joint measurement in the Bell basis using one qubit of the shared EPR pair
together with the qubit in the | i state; (3) Finally, Alice communicates the measurement
outcome to Bob (in the form of two classical bits s1 = w and s2 = v) who then applies
the corrections X v and Z w to his qubit of the EPR pair. A representation of the protocol
is depicted in Fig. 2.
Let |
0i
be the initial state of the teleportation circuit shown in Fig. 2. It is a three-
qubit state composed by a Bell state, |
are unknown amplitudes:
|
00 i,
and the state | i = ↵|0i + |1i, where ↵ and
1 h
p
i
⌘
↵|0i1 |00i + |11i
0
2
+ |1i1 |00i + |11i
23
23
i
(2.6)
Alice uses the qubit she wants to send to Bob as the control for a CNOT gate whose
target is her EPR qubit. The resulting state is the following:
|
1 i123
⌘ CN OT12 ⌦ I3 |
0 i123
1 h
= p ↵|0i1 |00i + |11i
2
23
+ |1i1 |10i + |01i
Next, she applies the Hadamard gate to the first qubit:
|
2 i123
⌘
H1 ⌦ I2 ⌦ I3 | 1 i123
i
1h
=
↵ |0i + |1i |00i + |11i + |0i |1i |10i + |01i
2
1h
=
|00i ↵|0i + |1i + |01i ↵|1i + |0i
2
i
+ |10i ↵|0i
|1i + |11i ↵|1i
|0i
23
i
(2.7)
(2.8)
(2.9)
(2.10)
Finally, Alice measures her qubits in the computational basis {|00i, |10i, |01i, |11i}, which
collapses Bob’s qubit into one of the following states (depending on Alice’s measurement
outcomes, represented by the bits w and v):
14
Alice measurement’s outcome (wv)
Bob’s resulting state
00
↵|0i + |1i
01
↵|1i + |0i
10
↵|0i
11
↵|1i
|1i
|0i
Therefore, Bob is able to recover Alice’s | i state in his lab by applying the correction
operator X v Z w to the qubit he possesses, where v, w 2 {0, 1} are the two classical bits
sent by Alice.
2.1.3
Teleportation as a quantum computing primitive
In [52], Gottesman and Chuang showed that the teleportation scheme can be used
as a computational primitive if the initial state used as resource state can be chosen
appropriately. By doing so, it is possible to teleport a quantum state “through” a logical
gate U , that is, instead of obtaining the final state Z w X v | i, one would obtain Z w X v U | i,
which, after the Pauli corrections, is exactly the quantum state one would obtain by
directly applying gate U to the initial state. In fact, this “trick” can be done for U of any
dimensionality (as long as the initial entangled state has the appropriate dimensionality),
giving as output states of the form P s U | i, where P is a collection of Pauli operators
depending on the measurements outcomes (jointly represented as s). In what follows we
show how to implement gates from the universal gate-set {Rz (✓), Rx (✓), CN OT }1 , using
the teleportation scheme as a primitive.
2.1.3.1
Teleporting a single-qubit gate
The implementation of single-qubit gates is achieved by a simple modification of the
measurement basis used in the original teleportation protocol. Instead of using the Bell
basis [formed by the states in Eq. (2.5)] to perform Alice’s joint measurement, we use a
different basis, called the “rotated Bell basis”:
| (U )vw i = U † ⌦ I|
vw i.
(2.11)
Now we re-do the calculation of the teleportation protocol circuit but using the rotated
basis in Eq. (2.11) instead of the original Bell basis. Projecting the state in Eq. (2.6)
1
where Rz (✓) and Rx (✓) are rotations by an angle ✓ around the X and Z axis, respectively.
15
onto the rotated Bell state | (U )00 i yields:
12 h
(U )00 ||
00 i123
=
12 h
(U )00 || i1 |
(2.12)
00 i23
i1 | 00 i23
h
00 | ↵U |0i1 |00i + |11i
i
+ U |1i1 |00i + |11i 23
⌘
1⇣
=
↵U |0i + U |1i
2
3
1
=
U | i3
2
=
(2.13)
12 h 00 ||U
1
= p 12 h
2
23
(2.14)
(2.15)
(2.16)
where I used ↵U |0i + U |1i ⌘ U (↵|0i + |1i). Note that the state in Eq. (2.16) is Alice’s
original state | i after the unitary U has been applied.
Thus, if Alice perform a measurement on her qubits using the rotated Bell basis in
Eq. (2.11), Bob’s state would be one of the following possibilities (depending on Alice’s
outcome):
Alice measurement’s outcome
00
01
10
11
Bob’s resulting state
↵U |0i + U |1i
↵U |1i + U |0i
↵U |0i
↵U |1i
U |1i
U |0i
Therefore, Bob is able to produce the state U | i in his lab by applying the correction
operator X v Z w to the qubit he possesses, where v, w 2 {0, 1} are the two classical bits
sent by Alice. Since an arbitrary SU(2) rotation can be decomposed as U (↵, , ) =
Rz ( )Rx ( )Rz (↵) (for some ↵, , ), arbitrary single-qubit unitaries can be implemented
by composing the teleportation protocol with rotated Bell bases, more specifically, with
unitaries U1 = Rz (✓) and U2 = Rx (✓).
2.1.3.2
The CNOT gate
Consider the scenario where two spatially separated qubits need to be teleported to
the same place. This can be achieved by simply considering a double teleporter (Fig 3),
that is, two separated teleportation protocols where the “Bob” parts of each protocol are
at the same place (or arbitrarily close to each other). A few comments about this protocol
are noteworthy. First of all, remember that the teleportation protocol does not destroy
16
|ai
|
s1
Bell
s2
00 i
|
00 i
Bell
s3
X
Z
|ai |ai
X
Z
|bi |bi
s4
|bi
Figure 3: The double teleporter: two teleportation protocol circuits (Fig. 2) combined to
teleport two spatially separated states |ai and |bi to the same place.
the correlations of the teleported qubits. It can be used to teleport both entangled and
mixed states.
Without loss of generality, suppose now that the double teleporter is used simply to
teleport two different single-qubit states |ai and |bi. A CN OT can be applied at the
end of the circuit, as is done in Fig. 4, and then the computation can be carried on.
The implementation of CN OT gates, together with the ability of implementing arbitrary
SU (2) rotations (Sec. 2.1.3.1), is enough to achieve universality in quantum computing.
However, if one wants to implement a CN OT gate using only resource state preparation, measurements and classically controlled operations - as is the case for single-qubit
gates (Sec. 2.1.3.1) - an adaptation of the double teleporter needs to be done. Starting
from Fig. 4, we move the CN OT gate backwards through the Pauli gates until it reaches
the beginning of the circuit. By doing so, the set of Pauli gates in the circuit changes to
a different set of gates (still Pauli gates), as depicted in Fig. 5. Therefore, if the state
| iabcd = CN OTbc |
00 iab | 00 icd
can be produced, a CN OT gate can be applied to any two
qubits using the double teleporter protocol.
The teleportation-based quantum computer uses adaptations of the quantum teleportation protocol to perform computations. Single-qubit unitaries are implemented by using
the rotated Bell basis (Eq. 2.11) instead of the standard Bell basis from the original tele-
17
|ai
|
|
s1
Bell
s2
00 i
00 i
X
Z
X
Z
s3
Bell
s4
|bi
Figure 4: A double teleporter with a CN OT gate being applied to the teleported states.
portation protocol. This procedure requires the production (and possibly storage) of |
00 i
states, the ability of performing measurements in the rotated Bell basis and classically
controlled Pauli gates, which application depend on the Bell measurements outcomes.
On the other hand, to implement a CN OT gate the state | iabcd = CN OTbc |
00 iab | 00 icd
needs to be produced to be used as resource state for the double teleporter protocol. Note
that both resource states, namely |
00 i
and | iabcd , can be produced offline, that is, before
and independently of the computation. Thus, the production of these states (remember
that the | i state can be produced by applying a CN OT gate to a pair of |
00 i
states) can
be probabilistic, such that only “well-produced”, high-fidelity resource states are stored to
be used in the computation.
In summary, using simple adaptations of the teleportation protocol [14], Gottesman
and Chuang showed [52] that quantum computation can be driven by measurements only,
performed on previously and independently prepared resource states. The teleportationbased model was the first measurement-based model for quantum computing; in the
following years the same key ingredients present in the model proposed by Gottesman and
Chuang were used to develop several other measurement-based models [95, 11, 73, 89].
18
|ai
|
s1
Bell
s2
00 i
X
| i
|
00 i
X
Bell
|bi
X
Z
Z
Z
s3
s4
Figure 5: Adapted version of the double teleporter: using the four-qubit state | iabcd , this
circuit applies the CN OT gate to the unknown states |ai and |bi using only measurements
and classically controlled operations.
2.2
The rise of measurement-based models
In the subsequent years, several different MBQC models were proposed. Michael
Nielsen [87] showed that universal quantum computing can be achieved if a quantum
memory is available and it is possible to perform projective measurements on up to four
qubits. In [73], Debbie Leung simplified this result showing that 2-qubit measurements
are enough to achieve universality in a teleportation-based quantum computation model.
Nevertheless, the model invented by Robert Raussendorf and Hans Briegel in 2001 [96],
the so called one-way quantum computation (1WQC) model, is arguably the simplest,
most elegant MBQC model.
There are at least two remarkable differences between the 1WQC model and the
other MBQC models. First, entangling measurements are no longer necessary in the
1WQC model: the computation is driven by single-qubit measurements alone, in contrast
with the need for entangling measurements in both Nielsen’s and Leung’s proposals. This
simplification is only possible because the model requires the preparation of a multi-qubit
highly-entangled state at the beginning, usually called the resource state, that scales with
the size of the computation to be performed. Within this framework, the role of entanglement in quantum computing is highlighted; Since all entanglement needed for the
19
computation is generated before the actual computation starts, identifying which entangled quantum states can be used as resource states in 1WQC has become an active area
of research.
A remark on terminology. Some authors refer to the one-way model as the cluster
state model, because the cluster state was the first proven universal resource for 1WQC.
In this thesis I refer to an MBQC model driven by single-qubit measurements as the oneway model, regardless of the resource state being used. This terminology is in agreement
with the spirit behind the name: it is called one-way model because the resource state is
“consumed” during the computation, possibly becoming completely useless after the end
of the computation. In contrast, in the MBQC models where entangling measurements
are allowed, even the measured qubits can be recycled and used again in the computation.
2.3
A brief description of the one-way QC model
In this section I give a brief description of the main concepts of the one-way model,
which will be explained in more depth in the following sections. The one-way model for
quantum computing requires the initial preparation of a multi-qubit highly entangled state
over which the information processing will be driven by single-qubit measurements. A
particularly simple resource state is the so-called cluster state, which can be constructed
by arranging qubits in a square lattice and allowing an Ising-type interaction between
neighboring qubits (see Fig. 6-a). The cluster state is said to be a universal resource2 for
1WQC because any quantum algorithm can be performed using a cluster state of sufficient
size.
Once the resource state is prepared, the information processing is driven by projective single-qubit measurements. Each time a measurement is performed, the measured
qubit becomes unentangled with respect to the remaining qubits in the resource state;
the measured qubit itself has no importance for the computation being implemented only the measurement outcome has - and therefore can be discarded. Depending on the
measurement outcome, the remaining resource state is projected onto one of two possible
states - each of which associated to one measurement outcome. If the resource state is
projected onto an ‘undesired’ state3 , a set of corrections must be applied to map it to the
2
A formal definition of universal resources for 1WQC is given later, in Def. 1 (Sec. 2.4).
At this point is hard to make clear what “undesired state” means in our context. For now, it suffices to
say that this is the scenario where the computation is not heading to the correct answer (hence “undesired
state”), and hence must be mapped back onto the right track. In Chapter 3 I explain in detail when and
3
20
‘correct’ state, before the next measurement takes place. Alternatively, the measurement
bases of the next measurements can be modified in order to counteract the side-effects
of previous measurements. This process continues until the algorithm is done and there
are no more measurements to be performed. Let us now analyze each step of the 1WQC
model more carefully:
• Resource state. There are several different quantum states that can be useful
resources for 1WQC. There are both ‘natural’ - arising as the ground state of a
Hamiltonian - and artificial resource states, which need to be constructed before
the actual computation begins. A specially useful resource state family is the so
called graph states [60]. Graph states can be created in two steps: (1) Preparation of n qubits in the |+i =
p1 (|0i
2
+ |1i) state and (2) application of the unitary
gate CZij = diag(1, 1, 1, 1) to selected pairs of qubits i and j. These states are
called graph states because a graph can be used to represent the physical system,
by associating each qubit to a vertex and each application of a CZ gate to an edge
in the graph. Moreover, there are sub-classes of graph states which are known to
be universal resources for 1WQC, in the sense that arbitrary quantum computation
can be performed using these states. The most well-known case of universal graph
state is the square lattice graph state, usually referred to as the cluster state. The
issue of universality of resource states will be discussed in Section 2.4.
• Measurements. The information processing is driven by projective measurements
performed using appropriately chosen bases, which depend on the resource state
over which the measurements will be performed and the computation we would like
to run. For graph states, the following bases are usually used:
⇢
|0ii + ei✓i |1ii |0ii ei✓i |1ii
p
p
Bi (✓i ) =
,
,
2
2
which for the sake of simplicity will be represented simply as Bi (✓i ) = {|+✓i i, |
(2.17)
✓i i}.
To represent the measurement result, we use the bit value s 2 {0, 1} as a label for
the basis eigenvectors; Hence, si = 0 means that qubit i was projected onto the |+✓i i
eigenstate whereas si = 1 is associated to |
✓i i.
The measured qubit itself is not
important for the computation and can therefore be discarded after the bit value si
is recorded. The measurement outcome tells us what is the state of the remaining
how an undesired state (obtained after a measurement have been performed) can be corrected, a problem
which is intimately related to the usefulness of a given resource state.
21
qubits in the resource state, where the computation is actually being performed. In
section 2.5 I explicitly describe the effect of measuring a graph state qubit in the
Bi (✓i ) = {|+✓i i, |
✓i i}
basis and discuss the role of measurements in the Pauli bases
✓ = 0 and ✓ = ⇡/2.
• Adaptiveness. Due to the randomness of the measurement results, it is not possible to predict the quantum state of the remaining qubits in the resource state. After
each measurement, the resource state collapses onto one of two possible states, each
of which associated to the measurement outcome s 2 {0, 1}. This randomness can
be accounted for by adapting the measurement angles of future measurements de-
pending on the outcome of previous ones. This concept of canceling the side-effects
of randomness on the resource state in order to perform deterministic computation
- a technique known as adaptive measurements - will be further detailed in Section
2.6.1.
• Output. The output of a 1WQC protocol can be either quantum or classical. In
the case where the goal is to prepare a specific quantum state, a subset of the cluster qubits is left unmeasured and some final unitary corrections must be applied
to counteract the random effect of previous measurements. On the other hand, in
the cases where some calculation is being performed - and hence a classical output
is expected - the output qubits are commonly measured in the computational basis {|0i, |1i} and the measurement results reinterpreted in the light of all previous
measurement outcomes that might have influenced the read-out measurements.
A one-way quantum computer can be seen as a classical computer which has access to
a quantum resource state. The classical computer has the algorithm stored in its memory
- a description of the order in which measurements must be done and the corresponding
measurement angles - as well as the results of all performed measurements. Moreover, the
classical computer needs to calculate whether a given measurement angle must be adapted
to counteract the randomness of previous measurements, changing the measurement angle
if necessary. In fact, this description of a one-way quantum computer makes even more
evident one of the main properties of the model: the clear separation between classical and
quantum computational resources. In contrast to the quantum circuit model, where the
whole information processing is driven by quantum operations, in the one-way quantum
22
(a)
(b)
(c)
(d)
Figure 6: Universal graph states: (a) Square Lattice, (b) Triangular Lattice, (c) Hexagonal
Lattice, (d) Kagome Lattice.
computer all ‘quantumness’ is in the resource state; the information processing is driven
by calculating some functions of the correlations between the measurements performed
over the resource state.
2.4
Resource states
A resource state is a multi-qubit highly entangled state used as a resource for a oneway quantum computer. In the 1WQC model, all entanglement that will be used in the
computation is available right at the beginning - no entangling operation needs to be
applied during the computation. The resource state can be prepared in several different
configurations, depending on the computational task to be performed. There are some
resource state families which enable any computation to be performed. We refer to those
families of states as universal resource state families for 1WQC. More formally, universal
resource state families can be defined as follow:
Definition 1 (Universal resource state families [108]) Let
finitely many states:
= {|
1 i, | 2 i, ...}.
The family of states
be a family of in-
is said to be a universal
resource state family for 1WQC if for each n-qubit state | i there exists a m-qubit (m
state | i 2
such that the transformation | i ! | i|+im
n
n)
can be implemented deter-
ministically by local operations and classical communication (LOCC).
That is, any quantum state | i can be prepared using only states within the family
. In
other words, any unitary operation U such that | i = U |+i can be implemented. Note
n
that here universality is a property of a family of states, and not of a single state4 .
4
For the sake of simplicity, however, we usually say that cluster states, for instance, are a universal
resource for 1WQC, although the precise statement would be that the cluster state family is a universal
resource state family for 1WQC.
23
The one-way model highlights the role of entanglement in quantum computing. Measurements on entangled states never increase the amount of entanglement and therefore,
in the one-way model the entanglement is “consumed” during the computation. Initially,
it was believed that a higher amount of entanglement would result in a greater usefulness
of the quantum state. However, Gross, Flammia and Eisert [56] showed that some quantum states are too entangled to be useful, i.e., that the correlation in the statistics of any
set of measurements over such states could be classically simulated. On the other hand,
several states are known not to have enough entanglement to be useful for 1WQC, as it
is the case for the 1-dimensional cluster state [88] and rectangular cluster states having
one side that scales logarithmically with respect to the other one [114].
A resource state can be artificially constructed or it can be a “natural” state, arising
as the ground state of a given Hamiltonian. A good example is the Affleck-Kennedy-LiebTasaki (AKLT) state [8], that first appeared in the context of condensed matter physics.
The AKLT state is the ground state of a two-body interaction Hamiltonian and it is
separated from the excited states by a finite energy gap. It was shown that 1-dimensional
AKLT chains can be used to perform simple quantum computing tasks [83] and that by
properly coupling many AKLT chains universal quantum computing can be achieved [19].
In [113], it was shown that two-dimensional AKLT state on a honeycomb lattice is also
a universal resource for measurement-based QC. Thus, AKLT states stand as one of the
most interesting resource states for 1WQC, due to the possibility of preparing it in solid
state systems just by cooling them.
Although two-dimensional cluster states can be used as a resource for universal quantum computation in the one-way model, arbitrary graph states may, or may not, serve
for the same purpose; investigating which kinds of entangled states are useful resources
for 1WQC is an important and active area of research [56, 108, 85, 112].
2.4.1
Cluster states and graph states
A particularly useful class of resource states are the so-called graph states [60]. A
graph state is the state of a multiqubit system that can be represented by a graph in the
following way: Each qubit of the system is represented by a vertex and the entanglement
between two qubits is represented by an edge. More formally, graph states can be defined
as follows.
Definition 2 (Graph states [60]). Let G = (V, E) be a graph, where V and E are the
24
sets of vertices and edges of G, respectively. A graph state |Gi is the unique simultaneous
eigenstate with eigenvalue +1 of the operators:
Ki = X i
Y
Zj ,
j2N (i)
(2.18)
8i 2 V
where N (i) stands for the set of vertices which are neighbors of i in G.
Cluster states are a particular type of graph states, where the graph is a square
lattice of some dimension d, as shown in Figure 6-a. Due to the restricted entanglement
geometry of cluster states, a given algorithm can in general be implemented using fewer
qubits in a graph state with more suitable geometry. However, there are several physical
implementations where the regular design of the cluster make it easier to be constructed.
Proposition 1 (Graph state preparation [100]). A graph state can be created by
preparing a collection of qubits in the |+i =
p1 (|0i
2
under the Hamiltonian:
H = ~g
X I (i)
(i,j)2EG
p
(i)
z
2
⌦
I (j)
p
(j)
z
2
= ~g
+ |1i) state and letting them evolve
X
(i,j)2EG
|1ii h1| ⌦ |1ij h1|
(2.19)
for a time T = ⇡g .
Proof. The unitary evolution driven by the Hamiltonian in Eq. (2.19) can be written as:
UH = exp(i
⇣
⌘
X
TH
) = exp i⇡
|1ii h1|⌦|1ij h1| =
~
ij
(i,j)2EG
Y
(i,j)2EG
⇣
⌘
exp i⇡|11ii,j h11|ij (2.20)
This is a product of the so-called controlled-phase gate, or simply CZ gate, and can be
more conviently written as:
CZij = |0ii h0| ⌦ Ij + |1ii h1| ⌦ Zj
(2.21)
25
which applies the Pauli Z operator (phase flip) to qubit j if qubit i is in state |1i. The
CZ gate satisfies the following properties:
CZij (Xi )CZij† = Xi Zj ,
(2.22)
CZij (Xj )CZij† = Zi Xj ,
(2.23)
CZij (Zi )CZij† = Zi ,
(2.24)
CZij (Zj )CZij† = Zj .
(2.25)
Now we are in the position to prove the proposition. Evolving the multi-qubit (separable) state |+i⌦n accordingly with UH yields:
UH ⌦i2VG |+i = UH
=
Y
Xu
u2VG
Y
=
u2VG
=
Y
u2VG
i2VG
CZij
(i,j)2EG
Y ⇣
O
Xu
Y
Xu
u2VG
Y
v2N (u)
(2.26)
|+i
Zv
O
i2VG
(2.27)
|+i
⌘ Y
CZij
(i,j)2EG
O
i2VG
|+i
(2.28)
(2.29)
Ku |Gi
where Eq. (2.18) was used in the last equality. Since Eq. (2.29) satisfies Definition 2, the
statement in this proposition holds.2
Let us now consider as an example the 1D cluster state with three qubits. The graph
state associated to this system can be constructed as follows:
|G3 i = CZ12 CZ23 |+i1 |+i2 |+i3
⇣
1
=
CZ12 |+i1 |00i23 + |01i23 + |10i23
2
1 ⇣
= p |000i + |100i + |010i |110i
2 2
⌘
+ |001i + |101i |011i + |111i
123
|11i23
⌘
(2.30)
(2.31)
(2.32)
(2.33)
26
It is easy to verify that this state satisfies the three eigenvalue equations:
X1 Z2 |G3 i = |G3 i
Z1 X2 Z3 |G3 i = |G3 i
Z2 X3 |G3 i = |G3 i
(2.34)
(2.35)
(2.36)
as required by definition 2. The graph state we just constructed is locally equivalent
to the well-known GHZ state [2], which is widely used in several quantum information
protocols. To see this, first apply the Hadamard gate to qubit 2 and then change the
basis from {|0i, |1i} to {|+i, | i}:
⌘
1⇣
|000i + |110i + |011i + |101i
2
⌘
1 ⇣
= p | + ++i + |
i
2
I1 ⌦ H2 ⌦ I3 |G3 i =
which is the GHZ state.
(2.37)
(2.38)
The first universal resource to be discovered was the two-dimensional cluster state.
Following this result, other 2D regular lattices were proven to be universal resources, such
as the triangular, hexagonal and Kagome lattices [108] (see Fig. 6). Moreover, any graph
from which one can obtain (by deleting some vertices) one of the aforementioned universal
lattices can also be considered as a universal resource (this is the case for some fractal
lattices [78]). This is so because a qubit can be easily removed from a graph state by
simply measuring it in the computational basis {|0i, |1i}, as we show in Section 2.5.2.
A non-universal graph state is not necessarily useless for one-way quantum computing.
There might exist some specific computational tasks for which non-universal graph states
can be used and still outperform classical computers. It is an active area of research to
identify which resource states are useful for 1WQC as well as which lead to measurement
statistics that can be efficiently simulated in a classical computer.
2.5
Measuring graph state qubits
In the last section we reviewed several different quantum states that can be used for
quantum information processing in the 1WQC model. Once the resource state has been
prepared, the quantum information processing is driven by projective measurements:
27
Definition 3 (Projective measurements [86]). A projective measurement is described
by an observable M , which is an Hermitian operator with spectral decomposition given by
M=
X
(2.39)
mPm
m
where Pm is the projector onto the eigenspace of M associated to the eigenvalue m. The
possible outcomes of the measurement are labelled by the eigenvalues of observable M . If
a quantum system in the | i state is measured, the probability of getting the outcome m
is given by
(2.40)
p(m) = h |Pm | i,
whereas the resulting state (in the case outcome m occurred) would be
p
1
p(m)
Moreover, the projection operators {Pm } must satisfy the completeness relation,
I, and the orthogonality relation, Pm Pm0 =
(2.41)
Pm | i.
mm0 Pm .
P
m
Pm =
Two remarks about nomenclature shall be made at this point. Most times I will refer
to the set of orthogonal projectors {Pm }, instead of the associated observable M , to
describe a projective measurement. Moreover, the phrase “measurement in the basis |mi”
(where {|mi} denotes an orthonormal basis) refers to the measurement of observable M ,
constructed according to Eq. (2.39) using the projectors Pm = |mihm|.
A simple but important example is the single-qubit measurement in the {|0i, |1i} basis
(i.e., the Pauli Z observable), which in quantum information is known as the computational basis. The projectors are P0 = |0ih0|, associated with the eigenvalue m = 0 and
P1 = |1ih1|, associated to m = 1. Suppose that the qubit is going to be measured is in
the | i = a|0i + b|1i state. Then, the probability of obtaining the measurement outcome
m = 0 is
p(0) = h ||0ih0|| i = |a|2 .
(2.42)
Similarly, the probability of obtaining the measurement result m = 1 is given by p(1) =
|b|2 .
In Section 2.5.2 we will see how measurement in the computational basis plays an
important role in computing using graph states. Before getting to that, however, I shall
introduce the measurement bases usually used in the 1WQC model (Section 2.5.1). Last
but not least, I analyze in Section 2.5.3 the role of measurement in the {|+i, | i} basis5 ,
5
That is, the measurement of the Pauli X observable.
28
that can be used to teleport logical qubits from one point to another in the graph.
2.5.1
Single-qubit measurements
Let us start by analyzing the effect of a single-qubit measurement in a simple graph
state. For reasons that will become clear later, we use the measurement bases in Eq.
(2.17), reproduced below:
Bi (✓i ) =
⇢
|0ii + ei✓i |1ii |0ii ei✓i |1ii
p
p
,
2
2
which can be represented simply as Bi (✓i ) = {|+✓i i, |
✓i i}.
(2.43)
,
Consider an input state
| i = ↵|0i + |1i and a resource state |Gi prepared in the following way:
(2.44)
|Gi = CZ12 | i1 |+i2
= CZ12
h 1
i
↵|0i + |1i 1 p |0i + |1i
2
2
1
= p ↵|00i + ↵|01i + |10i
2
Measuring the first qubit in the basis Bi (✓i ) = {|+✓i i, |
|11i
(2.45)
(2.46)
12
✓i i},
we obtain one out of
two possibilities. If qubit 1 is projected onto the |+✓ i state, the eigenstate associated to
h
i
1
i✓
i✓
p
|0i + ↵ e
|1i .
the +1 eigenvalue, qubit 2 is projected into the state 2 ↵ + e
On the other hand, if the measurement result is | ✓ i, qubit 2 is projected into the state
h
i
i✓
i✓
p1
↵
e
|0i
+
↵
+
e
|1i
. Thus, the state of qubit 2 can be represented as
2
X s HP (✓)| i,
(2.47)
where X is the Pauli operator, s is a bit s 2 {0, 1}, H is the Hadamard gate and
P (✓) = diag(1, ei✓ ) the phase gate. By definition, s = 0 is associated to the eigenstate
|+✓ i while s = 1 gives the eigenstate |
✓ i.
The Pauli operator X s , which depends on the
measurement result s, is called the by-product operator. In fact, by-product operators
can be composed by Pauli operators X and Z; they appear in the computation as a
consequence of the randomness of the results of individual measurements6 . The simple
example above shows how to teleport a quantum state through the gate:
!
1 1 ei✓
J(✓) = HP (✓) = p
2 1
ei✓
6
(2.48)
As we will see in Sec. 2.6.1, those by-product operators become troublesome when composing several
J gate teleportation protocols. We will see that they can be accounted for by adapting measurement
bases during the process of computation, according to the results of previously performed measurements.
29
using only a two-qubit graph state, one single-qubit measurement and a Pauli correction.
From now on, I will refer to the aforementioned protocol as the J-gate teleportation
protocol. As I explain in section 2.6.3, J(✓) gates play a central role in this model since
together with the CZ gate they constitute a universal gate-set for quantum computing.
2.5.2
Removing redundant qubits
When using universal graph states (e.g., a cluster state) to implement a 1WQC protocol that simulates several concatenated gates, it might happen that some qubits in
your resource state must be removed. For instance, suppose one wants to simulate the
implementation of a CN OT gate as in Figure 7-b but using a 3 ⇥ 3 cluster state; five
qubits would need to be removed in order to have the exact graph depicted in Figure 7-b.
Fortunately, qubits can be easily removed from graph states by simply measuring them
in the Z eigenbasis. This follows from the definition of graph states (Def. 2) and the pair
of identities:
|0ii h0|CZij = |0ii h0| ⌦ Ij
|1ii h1|CZij = |1ii h1| ⌦ Zj
(2.49)
(2.50)
Therefore, if the measurement outcome is si = 0, there is no effect on the remaining
qubits in the graph (Eq. 2.49); if si = 1, all qubits initially connected by an edge to the
measured qubit will gain a phase (Eq. 2.50), given by the Pauli Z gate. As will be shown
in Section 2.6.1, Pauli corrections can be handled by simply adapting the measurement
basis on that site. Therefore, the side effect of measuring a qubit in the Z basis can be
easily compensated.
Interestingly, measurements in the X and Y eigenbasis also result in new graph states:
they not only delete a vertex from the graph (as the Z measurement does) but also change
the geometry of the graph nearby the measured qubit. An overview of the effect of Pauli
measurements in graph states can be found in Appendix 1.
2.5.3
Moving the logical state of a qubit in a graph
Another particularly useful measurement basis is the obtained by fixing ✓ = 0 in Eq.
(2.17), i.e., measuring the Pauli X observable. Suppose the one needs to move a qubit
state from one part to another in the graph (similarly to the teleportation protocol).
There are several situations where that might be necessary (or at least convenient), as
30
0
0
0
0
(a)
(b)
0
1
1
1
0
3
3
3
(c)
0
2
2
2
4
4
4
(d)
Figure 7: Graphs implementing (a) Arbitrary single-qubit unitary, (b) CN OT gate, (c)
CZ gate, (d) two arbitrary single-qubit rotations followed by a CZ gate and then other
two single-qubit rotations.
for example in a graph with such a geometry that a two-qubit gate can be simulated just
in specific regions of that graph. Hence, we would frequently need to move qubit states
around so that two-qubit gates can be applied to any pair of them.
Suppose we need to move a qubit state (in state, say, | i) from a given vertex i to
another j. First, we need to build a “wire” with an even number of qubits linking the
two vertices; this can be done by deleting the qubits in the neighborhood of the wire
(using measurements in the Z basis), resulting in a wire of qubits connected by edges
as depicted in Fig. 7-a, for instance. Then, we measure all qubits in the wire in the X
basis. Each measurement will teleport the state to the next vertex while applying the
gate J(✓ = 0) = H, as explained in Section 2.5.1. Since H N = 1 (for N even), by the end
of the process we will have moved the qubit state to vertex j, as desired.
2.6
Simulating the circuit model
The 1WQC model is equivalent to other quantum computing models. For instance,
arbitrary quantum circuits can be efficiently simulated using the one-way model if a
graph state of sufficient size is available and arbitrary single-qubit measurements can
be performed. In this section I explicitly show: (i) how to simulate arbitrary quantum
circuits. To do so, it is sufficient to show how to simulate any gate of an universal gateset; and (ii) that the subgraphs simulating each of those gates can be “glued” together,
allowing the concatenation of arbitrary gates.
It turns out that the gate simulated by measuring a graph state qubit in the B(✓)
bases (as in the J gate teleportation shown in Section 2.5.1), namely J(✓) = HP (✓),
together with the CZ gate, constitute an universal gate-set for quantum computing [34].
In the following sections I show how to concatenate many J-gate teleportation protocols
in order to simulate any given quantum circuit.
31
2.6.1
Universal single-qubit rotations and the need for adaptive
measurements
Here I show how to compose the J-gate teleportation protocol shown in Section 2.5.1.
As an useful example, we show how to perform arbitrary single-qubit SU (2) rotations.
Using the Euler decomposition, we can write such unitary as
(2.51)
U (↵, , ) = Rz ( )Rx ( )Rz (↵)
for some angles , , ↵. We can rewrite Eq. (2.51) in terms of Eq. (2.48) (the J gate) by
using the identities Rz ( ) = J(0)J( ) and Rx ( ) = J( )J(0). Thus, an arbitrary rotation
U 2 SU (2) can be written using only J gates:
(2.52)
U (↵, , ) = J(0)J( )J( )J(↵).
In the previous section, we have seen how to apply a J gate to an arbitrary state | i using
only single-qubit measurements on a Bell pair and classical processing. The question that
naturally arises is the following: Could one apply the unitary in Eq. (2.52) by composing
the J gate teleportation protocol several times? This is indeed possible, but it involves
some issues that we now analyze. First, consider the following procedure. Apply the
J gate teleportation protocol to a state |Input1 i = |
0i
but instead of measuring the
output qubit in the computational basis (and doing some classical post-processing to
account for the X s factor), we feed it to the protocol again, resulting in the output state
|Output2 i = X s2 J(✓2 )X s1 J(✓1 )|
0 i.
Repeating this process two more times, we end up
with the following description of the final state:
X s4 J(0)X s3 J(✓3 )X s2 J(✓2 )X s1 J(✓1 )|
0i
(2.53)
which means that the desired unitary implementation occurs only when s1 = s2 = s3 = 0
(the X s4 can be accounted for by reinterpreting the read-out measurement, performed in
the computational basis). Therefore, the X s operators in Eq. (2.53) (known as by-product
operators), which can easily be dealt with when performing a single application of the J
gate teleportation protocol, become a troublesome factor when several of those protocols
are composed.
One could think about solving this problem by evolving the output qubit of each J gate
teleportation protocol (whenever s = 1) according to to a Hamiltonian that implements
a X gate, before feeding it as the input of the next protocol. By doing so, the final state
32
|Ini
|+i
|+i
|+i
|+i
|Ini
|+i
|+i
|+i
|+i
X
X
X
X
(a)
(b)
Figure 8: (a) composed J gate teleportation protocol and (b) 1WQC protocol using
adaptive measurements to implement the same unitary as in (a).
would be:
X s4 J(0)J(✓3 )J(✓2 )J(✓1 )|
(2.54)
0i
which applies the desired unitary. Although this procedure gives the expected result, it
can no longer be considered a 1WQC protocol. In one-way quantum computation, the
information processing is driven by single-qubit measurements only and therefore unitary
evolutions (to apply X) are not allowed during the computation. Interestingly, a genuine
1WQC protocol can be constructed from the “composed protocol” above, if we allow some
measurement angles to be conditioned to previous measurements outcomes. In what
follows I show how to use this conditioned choice of measurement angles to cancel out the
by-product operators of previous measurements; this technique is usually referred to as
adaptive measurements.
First, note that the measurement on qubit i commutes with gate CZjk for j, k 6= i.
Therefore, as long as the measurements are performed in sequence (from qubit 1 to 4), all
CZ gates used in the composed protocol can be applied before any measurement takes
place. By doing so, a 5-qubit entangled state is prepared before the computation begins, as
required in the 1WQC model. Next, we need to remove the X s operators from in-between
the J gates in Eq. (2.53). Using the the identities J(✓)X = ZJ( ✓) and J(✓)Z = XJ(✓),
Eq. (2.53) can be rewritten as
X s2 +s4 Z s1 +s3 J(0)J[( 1)s2 ✓3 ]J[( 1)s1 ✓2 ]J(✓1 )|
0i
(2.55)
which is no longer the implementation of a deterministic unitary, since only if s1 =
s2 = 0 is the implemented unitary the desired one. This can be dealt with by allowing
adaptive measurements, that is, the ability to change the angles depending on previous
measurement outcomes:
✓1 = ↵,
✓2 = ( 1)s1 ,
✓3 = ( 1)s2
(2.56)
33
Thus, combining Eqs. (2.55) and (2.56) we have exactly Eq. (2.51), as desired. Therefore,
adaptive measurements can be used to deal with the by-product operators in the middle
of a computation, allowing one to apply a given unitary deterministically. This subject
will be further explored in Chapter 3, where I address the question of which graph states
allows for deterministic 1WQC.
2.6.2
Two-qubit gates
The simulation of a CN OT gate can be implemented using the simple 4-qubit graph
shown in Figure 7-b. Consider the following labels for the graph in Figure 7-b: from left
to right, the top vertices are labelled by numbers 1, 2 and 3; the bottom vertex is labelled
by number 4. Let qubits 1 and 4 encode the control and target input qubits respectively,
while qubits 3 and 4 encode the control and target output qubits, respectively. Note that
in this compact implementation of the CN OT gate, qubit 4 encodes both the input target
qubit (before the measurements) and the output target qubit (after measurements). Let
|
1i
= a|0i + b|1i be the initial state of the target qubit and |
2i
= c|0i + d|1i the initial
state of the control qubit. The graph state is constructed in the following way:
|GT i = CZ24 CZ23 CZ12 |
(2.57)
1 i1 |+i2 |+i3 | 2 i4
= CZ24 CZ23 a|0i1 |+i2 + b|1i1 | i2 |+i3 | 2 i4
h
i
= CZ24 a|0i1 |0i2 |+i3 + |1i2 | i3 + b|1i1 |0i2 |+i3 |1i2 | i3 |
h
i
= a|0i1 |0i2 |+i3 c|0i + d|1i 4 + |1i2 | i3 c|0i d|1i 4
h
i
+ b|1i1 |0i2 |+i3 c|0i + d|1i 4 |1i2 | i3 c|0i d|1i 4
(2.58)
2 i4 (2.59)
(2.60)
(2.61)
The protocol consists in measuring qubits 1 and 2 on the |±i basis and correcting qubits
3 and 4 in accordance with the measurement outcomes s1 , s2 2 {0, 1}. For instance, if
s1 = s2 = 0, which means that qubits 1 and 2 collapsed onto the |+i state, the unmeasured
qubits are in the joint state
| i34 = h+|1 h+|2 |GT i1234
⇣
= a |+i3 | 2 i4 + |+i3 Z4 |
2 i4
⌘
⇣
+ b |+i3 |
2 i4
= ac|00i34 + ad|11i34 + bc|10i34 + bd|01i34
= CN OT43 |
2 i4 | 1 i3 ,
|+i3 Z4 |
2 i4
⌘
(2.62)
(2.63)
(2.64)
(2.65)
where the normalization factors were omitted. Note that the information on the target
qubit has been transferred from physical qubit 1 to 3 while the control input and output
34
state is encoded in the same physical qubit, namely qubit 4. Analyzing the other three
possible measurement outcomes (s1 , s2 ) 2 {(0, 1), (1, 0), (1, 1)}, it is easy to check that
the output state |
transformation:
out i
relates to the initial state |
|
out i
in i
= Xts2 Zts1 Zcs1 CN OT |
= |
in i
1 it | 2 ic
via the following
(2.66)
which applies a CN OT gate to the input state once the Pauli corrections are applied.
Since the entangling gate in the 1WQC framework we are using is the CZ gate, the
simulation of a CZ gate comes “for free”. In the preparation of the graph state, the CZ
gate is applied to several pairs of qubits (the choice of pairs depends on the geometry of
the graph) and are represented as edges in the graph. It turns out that those edges can
be used at any point of the computation to simulate the application of a CZ gate to the
qubit states in the vertices connected by the edges. In Chapter 3 I explain the difference
between using a graph edge to implement a CZ gate or a J gate teleportation protocol.
Consider the graph in Figure 7-c, where qubits 1 and 2 are input and output qubits at
the same time. It is trivial to see that this simple graph applies the CZ gate to the input
state. In fact, that graph can be composed with graphs implementing other unitaries. See
for instance Figure 7-d where the composition of the graph implementing the CZ gate
(Fig. 7-c) is composed with four graphs able to implement arbitrary single-qubit rotations
(Fig. 7-a).
2.6.3
Simulating arbitrary quantum circuits
The one way model is an universal model for quantum computation. In the last sections I have shown how to simulate both arbitrary single-qubit rotations and the CN OT
gate, which combined together constitute an universal gate-set for quantum computing.
Moreover, the simple gate in Eq. (2.48), that can be implemented (up to a Pauli correction) using only one single-qubit measurement, together with the CZ gate also constitute
a universal gate-set [34].
As an example, Figure 9 shows how a circuit is simulated in a cluster state. In the
figure, the input state is encoded in the qubits at the left-most column and the output in
the right-most one. First, the qubits that will not be used to simulate any gate from the
circuit are removed from the cluster. This can be done by measuring them in the Z basis
(as explained in Section 2.5.2); the removed qubits are represented as black dots in the
figure. After that, the remaining graph has the right geometry to simulate each of the
35
Time
Figure 9: The simulation of a quantum circuit in a cluster state. The black dots represent
qubits measured in the Z basis (and hence deleted from the cluster) and the circles with
arrows represent qubits measured in the B(✓) basis. Different arrow directions indicate
different angles of measurement. Each set of qubits boxed with a dashed line represents
the simulation of a given circuit gate.
gates from the circuit that will be simulated. There are four horizontal wires (simulating
the circuit wires) and some vertical lines connecting them (simulating a two-qubit gate
between the wires). Moreover, some pairs of qubits are X measured to move the output
state of a gate simulator building block to be the input state of the next one, as explained
in Section 2.5.3.
In fact, the universality of a measurement-based quantum computing model depends
basically on two properties: The resource state and the measurement bases. When a
resource state is said to be universal for QC, it means that there exist measurement
bases such that the statement holds. For instance, some of the known universal graph
geometries are: cluster state (X
a) [94], Brickwork state (X
Y plane measurements + Z measurements; see Fig. 6-
Y plane measurements only) [22], Hexagonal lattice (X
Y
plane measurements + Z measurements; see Fig. 6-c) [108], triangular lattice (X
Z
plane measurements only; see Fig. 6-b) [82].
36
2.6.4
Beyond quantum circuit simulations
In the last sections I showed how the cluster state can be used as a ‘canvas’ over which
a quantum circuit is simulated: after deleting some specific qubits from the cluster, the
resulting graph geometry is nothing but a composition of small building blocks, each of
which simulates a logical gate. However, the one-way model is not just a quantum circuit
simulator. In fact, many quantum algorithms can be implemented in a more compact
way when the one-way quantum computer is not simulating quantum circuits.
The circuit model encompasses the Schrodinger picture of Quantum Mechanics where
the quantum state changes by the action of well-defined unitary evolutions. A logical gate
in the circuit model is nothing but a unitary transformation in the Schrodinger picture
of quantum mechanics. On the other hand, in the one-way model there is no notion of
gates being applied nor unitary evolutions acting over a physical system. Therefore, the
one-way model introduces a completely different way of thinking about computation when
compared to the circuit model, since no logical gate is actually required (although it can
be understood using gates).
For instance, there are some optimization techniques in the one-way model which are
able to reduce the number of computational steps needed to simulate a quantum circuit in
such a way that the ‘gate building blocks’ can no longer be identified. That is, although
the new algorithm still implements the same computation, it can no longer be understood
as a concatenation of small logical gates (as in the circuit model). The one-to-one correspondence between measurements performed and single-qubit gates implemented shown
in previous examples (Section 2.6.1) is not a property shared by every one-way quantum
computation. Nevertheless, to design algorithms which are not built by composing simple
logic gates may represent a considerably difficult task.
2.6.5
Simulating quantum circuits with unbounded fan-out
The computational power of one-way quantum computation is equivalent to the power
of quantum circuits with unbounded fan-out [25]. This model of computation consists of
a generalization of the circuit model, where any two commuting gates can be performed
simultaneously, thus reducing the total number of computational time slices required to
implement a given computation. First explored by Høyer and Špalek [62], the unbounded
fan-out model is surprisingly powerful: the quantum part of Shor’s algorithm [105], for
instance, can be done in one single computational step in this model.
37
The equivalence between the computational power of 1WQC and the unbounded fanout model is, however, up to classical parity computations, necessary to calculate the
dependency between measurement outcomes and future measurement bases in the 1WQC
model. The extra power of the one-way model over the circuit model comes from its classical/quantum hybrid nature; this separation can be exploited to decrease the quantum
requirements of a given algorithm by increasing its classical processing. Although this
is not always possible, in Section 3.3 we explore some 1WQC optimizations techniques
which use the aforementioned classical/quantum separation to reduce the number of computational steps of a given algorithm. Moreover, in Chapter 6, I show what happens when
a certain family of optimized one-way quantum computations are translated back to the
quantum circuit model, in an attempt to optimize quantum circuits by back-and-forth
translation to the 1WQC model.
2.7
Clifford computations
In the 1WQC model, all measurements commute as quantum operations, since only
single-qubit measurements are allowed and qubits are discarded after being measured.
Therefore, if there is no need for adaptive choices of bases, all measurements can be
performed at once in the beginning of the computation. As a matter of fact, there is a
group of gates that can indeed be implemented independently of any other measurement
outcome in an one-way quantum computation - the so-called Clifford gates. The Clifford
group is generated by the CZ, H and Rz (⇡/2) gates, which can all be implemented in
the 1WQC model using Pauli measurements alone [more specifically, ✓ = 0 and ✓ = ⇡/2
in Eq. (2.17)]. Therefore, any Clifford group operation can be implemented using Pauli
measurements.
2.7.1
Pauli measurements and the Clifford group
The Pauli group Pn on n qubits is defined as the the group generated by the multiplication of n-fold tensor products of ±I, ±iI, X and Z. For instance, the following
operators are elements of the group P3 :
iX ⌦ i ⌦ Y , I ⌦
iY ⌦ Z, etc. A Clifford
operation C on n qubits is an operation such that for any P 2 Pn there exists a P 0 2 Pn
such that:
P C = CP 0
(2.67)
38
Hence, a Clifford operator can “pass through” a Pauli operator unaltered - just the Pauli
operator changes - and therefore no adaptive measurement is necessary to apply such
operations in the 1WQC model. In Sec. 2.6.1, I showed how adaptive measurements can
be used to counteract the by-product operator X after each measurement. However, due
to the Clifford gates property captured by Eq. (2.67), the by-product Pauli operators can
be handled without making use of adaptive measurements. More specifically, suppose an
equation similar to Eq. (2.53) but with Clifford operators instead of J gates:
X s4 C 4 X s3 C 3 X s2 C 2 X s1 C 1 |
0i
(2.68)
Using Eq. (2.67) the equation above can be rewritten as:
P s C4 C3 C2 C1 |
0i
(2.69)
where P is a Pauli operator and s = f (s1 , s2 , s3 , s4 ). The difference between Eqs. (2.69)
and (2.55) is the fact that the desired unitary is implemented deterministically in Eq.
(2.69), with no need for adaptive measurements. This nice property allows all measurements implementing a Clifford operator to be performed at once at the beginning of the
computation. Even if a Clifford operation appears in the middle of a quantum circuit,
the measurements that simulate those operations in the corresponding 1WQC protocol
can be done at the beginning of the computation.
With such a nice property, the question of how frequently Clifford operations appear
in useful QIP protocols naturally arises. As a matter of fact, Clifford gates appear in
virtually every quantum algorithm or protocol, such as quantum error-correcting codes
[3], Quantum Fourier Transform (QFT) [86] and the teleportation protocol described in
Section 2.1.2. Thus, the way the 1WQC model computes Clifford gates may provide
insights that otherwise would be difficult to see in the quantum circuit model.
2.7.2
The Gottesman-Knill theorem
Arguably the most important result regarding Clifford computation is the GottesmanKnill theorem:
Theorem 1 (Gottesman-Knill Theorem [86, 3]). A quantum computation involving
only (a) state preparation in the computational basis, (b) gates in the Clifford group, (c)
measurements of observables in the Pauli bases and (d) classically controlled operations
depending on previous measurement outcomes can be efficiently simulated in a classical
39
computer.
The original classical algorithm to simulate such circuits requires O(n3 m) computational
steps, where n is the number of initial qubits and m is the number of steps the quantum
computation requires in the circuit model. This result was further improved by Aaronson
and Gottesman in [6], reducing the number of computational steps to O(n2 ).
At first glance, one may think that since Clifford circuits can be efficiently simulated,
there is no advantage coming from the fact that all Clifford circuits can be implemented
in one single step in a one-way quantum computer. That is not exactly the case. Note
that the classical simulation algorithm is for circuits composed only by Clifford gates, and
not for a part of a generic quantum circuit composed only by Clifford group operations.
On the other hand, since Pauli measurements correspond to the implementation of gates
from the Clifford group, all Clifford operations can be reduced to classical processing in
the 1WQC model, regardless of in which point of the quantum computation they occur.
In this sense, it can be considered a stronger result than the Gottesman-knill theorem,
since it provides a reduction of the quantum processing even when the circuit is composed
just partially by Clifford gates.
2.7.3
Quantum-classical tradeoff
The single-step implementation of all Clifford operations in 1WQC does not come
entirely for free. Suppose we want to simulate in a one-way computer a quantum circuit composed by both Clifford and non-Clifford operations. After the round of parallel
Pauli measurements implementing all Clifford gates in the circuit, the exponent of the
by-product operators [P s in Eq. (2.67)] must be calculated before continuing with the
implementation of the non-Clifford part. However, to calculate the exponent s (which is
a modulo 2 sum) a classical computer requires log(k) computational steps, where k is the
number of (Pauli) measurements outcomes over which the exponent s depends upon.
Therefore, the simulation of a quantum circuit in 1WQC might not only reduce de
number of total computational steps, but also trade quantum requirements for classical
ones. That is, it takes a fully quantum algorithm and separates it into a quantum and
classical parts, conveniently “minimizing” the quantum part. Since quantum and classical
resources are usually regarded as incomparable resources7 , this separation of resources
can be regarded as one of the main interesting properties of the 1WQC model.
7
At least for technological reasons, since classical computation can be easily implemented nowadays.
40
It is worth noting that, regardless of the number of computational steps required,
the simulation of Clifford circuits in the 1WQC model is fundamentally different from
its simulation using the Gottesman and Knill (Theorem 1) approach: while the former is
simulating the quantum process itself, giving a quantum state as output, the latter just
simulates the measurement results, giving a purely classical output.
41
3
Deterministic computation using
quantum measurements
In this chapter we introduce the concept of deterministic computation in the oneway model. When a quantum algorithm is implemented, it has a finite probability of
success that is inherent to its construction. In the one-way model, where the information
processing is driven by measurements, a central question is to determine whether the
probabilistic character of quantum measurements can be compensated, such that it does
not aggregate error to the algorithm being implemented.
In fact, there are several different strategies to compensate the side-effect of measuring
an entangled qubit, all based in the same principle. In Sec. 2.5.1, I showed how to do so
for the simple example where two qubits are entangled by a CZ gate and then one of them
is measured in the B(✓) basis. In that case, it was enough to apply the Pauli operator
X s to the unmeasured qubit, where s labels the measurement outcome, to guarantee
the deterministic application of the unitary U = HP (✓). In this chapter I show how to
construct correction strategies for arbitrary graphs (whenever possible), in order to use
the associated graph states to implement deterministic computation.
As explained in Chapter 2, in this thesis I consider only single-qubit measurements
in the B(✓) basis applied over graph states. As a consequence of this (imposed) limitation1 , the side computer required to calculate how to adapt the measurement angles is
also limited: the only calculation that needs to be performed are modulo 2 summations.
A computer able to realize such summations, the so-called parity computer 2 , is considerably less powerful than a full-fledged classical computer. Extensions of this model were
discussed in [54, 55, 10], where the need of a more powerful computer (such as a classical
computer) appears in the model as a way to deal with the more elaborate resource state
1
Note that the original proposal of the one-way model has exactly the same limitation. Therefore,
even with those restrictions, universal quantum computing can be achieved.
2
The parity computer is a device implementing circuits composed only by CN OT and N OT gates.
The parity computer can solve a number of problems efficiently, such as simulating Clifford circuits
(Gottesman-Knill theorem, see Sec. 2.7.2), and calculating the parity of bit-strings.
42
and/or allowed measurements.
This chapter is divided into three parts. First, in Section 3.1, I review the so-called
measurement calculus [35], a formal language developed to work with 1WQC algorithms.
In this framework every 1WQC algorithm is represented by a sequence of commands
called a measurement pattern, which gives a description of the graph state to be created
and the program (measurement bases and temporal order of operations). The composability property of measurement patterns allows the construction of 1WQC algorithms
starting directly from their description in the circuit model; using a technique known as
standardization (Sec. 3.1.4), the measurement pattern obtained by composition of small
building blocks can be rewritten into a semantically equivalent form that specifies the
graph state that needs to be created to perform the computation in the 1WQC model. In
Section 3.2, I explain how a computation driven by quantum measurements can be made
deterministic. I explain in detail what the program (measurement pattern) is doing to
keep the computation deterministic and explain the different notions of determinism that
arise in the one-way model.
In the last part, Sec. 3.3, I review several correcting strategies generically known as
flows. A flow is a set of conditions over a graph that verify whether there exists a partial
time ordering to perform the measurements such that a deterministic computation can
be implemented (via adaptive measurements); when it exists, the flow also tells us how to
adapt the measurement bases to achieve determinism. The notion of flow is different from
the one of universality, discussed in Sec. 2.4. As we have seen in Def. 1, universality is
defined for a family of quantum states (for instance, the family of cluster states) while one
can test the flow conditions for a single graph state to see if deterministic computation
(even very simple ones) can be implemented using it as a resource state. Note also that
each state in a universal family of graph states satisfies the flow conditions; if it is universal
it certainly implements deterministic computation. I will give a more detailed explanation
on the importance of flows for 1WQC in Sec. 3.3.1.
I review three different types of flow: (1) Regular Flow 3 (regflow, for short) [33], (2)
Generalized Flow [24] and (3) Maximally-delayed Generalized Flow [81]. Those properties
are not mutually exclusive and therefore a graph can have two or more types of flow. A
comparison between all the aforementioned flows (together with some examples) concludes
the chapter.
3
Known in the literature simply by Flow [33]. To avoid confusion, in this thesis I will refer to that type
of flow as Regular Flow, letting the term flow to be used more freely to denote any flow-like correcting
strategy.
43
3.1
The measurement calculus
In this section, I review the very useful formal language known as measurement calculus [35], developed to study MBQC. I start with a description of the commands used in the
measurement calculus (Sec. 3.1.1), followed by the definition and main properties of measurement patterns (Sec. 3.1.2). Finally, I discuss two measurement calculus techniques
known as standardization (Secs. 3.1.3 and 3.1.4) and signal-shifting (Sec. 5.3.1).
In this section I am going to assume that all measurement patterns given - corresponding to algorithms in the 1WQC model - are perfectly runnable (i.e., physically
implementable, with no anachronical operation), so that we need not to be concerned
about determinism. In the following sections I show how to develop a correction strategy that allows deterministic computation (Section 3.2) and explore some of the most
well-known strategies (Section 3.3).
3.1.1
Commands
The measurement calculus language consists of four types of commands. The first is
the qubit initialization command Ni that prepares qubit i in the state |+i. The input
qubits are already given as an arbitrary state. The entangling command Eij ⌘ CZij cor-
responds to applying the CZ gate between qubits i and j. The single-qubit measurement
command Mi✓ corresponds to a measurement of qubit i in the basis Bi (✓) = {|+✓ i, |
with outcome si = 0 associated with |+✓ i, and outcome 1 with |
✓ i.
✓ i},
The measurement
outcomes si are sometimes referred to as signals. The corrections may be of two types,
either Pauli X or Pauli Z, and their application may be conditioned on the measurement
result of several previously performed measurements. Those dependencies are denoted
L
by si = j2J sj (with sj 2 {0, 1}), where J is the set of measurements upon which the
choice of basis for the measurement on qubit i depends on4 . Therefore, the conditioned
application of a Pauli Xi or Zi are represented as Xis and Zis , respectively, where the
application of the Pauli gate depends on whether s equals zero (Z 0 = X 0 = I) or one.
The table below summarizes the functionality of each command:
4
In the one-way model it is enough to use summation modulo 2 to account for all the dependencies.
There are other M BQC models which require a more powerful computer to calculate the dependency
net of the computation, as it is the case for the models introduced in [54] where a full-fledged classical
computer may be required.
44
Command
Functionality
Additional information
Ni
Prepares qubit i in the |+i state.
Ancila preparation; represented as
(unboxed) vertices in the graph.
Applies the CZ gate between Entanglement
Eij
Mi✓
X s and Z s
operator;
repre-
qubits i and j.
sented as edges in the graph.
Measures qubit i in the basis
The measurement outcome si = 0
Bi (✓) = {|+✓ i, |
is associated with |+✓ i, and out-
✓ i}
come 1 with |
✓i
Applies Pauli correction X (or The signal s can be the sum (modequivalently Z) depending on the
ulo 2) of many previously obtained
parity of the signal s.
signals.
A characteristic of the 1WQC model is that the choice of measurement bases may
depend on earlier measurement outcomes. These dependent measurements can be conveniently written as t [Mi✓ ]s , where
✓ s
t [Mi ]
( 1)s ✓+t⇡
⌘ Mi✓ Xis Zit = Mi
,
(3.1)
where it is understood that the operations are performed in the order from right to left
in the sequence. The t and s dependencies of the measurement Mi are called its Z and
X dependencies, respectively. Equation (3.1) can be intuitively understood by noting
that, since we are dealing only with measurements in the equator of the Bloch sphere, the
action of operators X and Z have straightforward consequences: The X operator flips
the computational basis (i.e., |0i ! |1i and |1i ! |0i), while Z adds a “-1” phase to the
eigenvector |1i (i.e., |0i ! |0i and |1i !
|1i). Thus, measuring a qubit in the B(✓) basis
after the application of operators X s Z t is the same as measuring it in the B[( 1)s ✓ + t⇡].
3.1.2
Measurement patterns
A measurement pattern P, or simply a pattern P, is defined as a set of measurement
calculus commands acting over open graph states, that is, quantum states associated to
the so-called open graphs:
Definition 4 (open graph) An open graph is a triplet (G, I, O), where G = (V, E) is a
undirected graph, and I, O ✓ V are the sets of, respectively, input and output vertices.
An open graph can be obtained from a graph G = (V, E) by choosing from the set of
working qubits V a subset of input qubits (I) and another subset of output qubits (O).
45
Examples of open graphs are shown in Fig. 10. Note that not all measurement patterns are
associated to physically sound sequences of operations. For example, if a given command
Aj in a pattern P has its application conditioned on a measurement outcome si , then the
measurement on qubit i must happen before command Aj in P. In this thesis we will be
concerned mostly with patterns that respect a few conditions of this sort, called runnable
patterns.
Definition 5 (runnable patterns) A measurement pattern is runnable, that is, corresponds to a physically sound sequence of operations, if it satisfies the following requirements:
• No command depends on outcomes of measurements that have not yet been performed;
• No command acts on a qubit already measured or not yet prepared, with the obvious
exception of the preparation commands;
• A qubit undergoes measurement (preparation) iff it is not an output (input) qubit.
As an example, take the pattern consisting of the choices V = {1, 2}, I = {1}, O = {2}
and the sequence of commands:
J(✓) := X2s1 M1 ✓ E12 N2 .
(3.2)
This sequence of operations does the following: first it initialises the output qubit 2 in
state |+i; then it applies CZ on qubits 1 and 2; followed by a measurement of input qubit
1 onto the basis B(✓). If the result is |
✓i i,
then the one-bit outcome is s1 = 1 and there
is a correction on the second qubit (X21 = X2 ), otherwise no correction is necessary. This
algorithm, compactly represented in Eq. (3.2), is the same analyzed in Section 2.5.1 and,
as we now know, it implements the unitary U = HP (✓) = J(✓). Since I am going to use
the measurement pattern in Eq. (3.2) extensively throughout this thesis, from this point
on I will refer to it as the J-gate pattern.
Next I review the definition of the quantum depth of a measurement pattern:
Definition 6 (quantum depth [25]) For a given open graph G = (V, I, O) and associated pattern P, we define the quantum depth of P, or simply depth(P), as the longest
subsequence (px ) of P, s. t. for any x, dom(px )\dom(px+1 ) 6= ;, where dom(Eij ) = {i, j},
46
dom(Xi ) = dom(Zi ) = {i} [ {j s.t. sj appears in }, dom(Mi✓ ) = {i}, dom(t [Mi✓ ]s ) =
{i} [ {j s.t. sj appears in t or s}.
Example: the measurement pattern P = X3s1 +s2 M2✓2 X2s1 M10 E23 E12 E13 associated to the
graph G = ({1, 2, 3}, {1}, {3}) has depth(P) = 6, since the longest dependent subsequence
of P is X3s1 +s2 M2✓2 X2s1 M10 E12 E13 .
Note that the quantum depth does not take into account the depth of the classical
side-processing coming from the summation mod 2 in s and t. This is so because it is
taken as assumption that classical computation is free, since it is much less demanding (in
terms of physical implementation) than implementing the quantum part of the algorithm.
From this part on, I will refer to the quantum depth of a pattern simply as the depth of
a pattern.
Measurement patterns can be combined into larger patterns. One way of combining
patterns is by composition, where the output of a pattern serves as input for the next
one. A pattern P1 can be composed with another pattern P2 if V1 \ V2 = I20 = O10 , where
I20 ✓ I2 and O10 ✓ O1 . This is the combination used to concatenate the MBQC simulation
of logical gates.
Definition 7 Two patterns P1 and P2 can be composed if V1 \ V2 = I20 = O10 , where
I20 ✓ I2 and O10 ✓ O1 , giving as result a pattern P12 = P1 P2 formed by the concatenation
of the commands in P1 and P2 and computational space given by:
V12 = V1 [ V2 ,
I12 = I1 [ I200 ,
O12 = O2 [ O100
(3.3)
where I200 = I2 /I20 and O100 = O1 /O10 .
Composition allows, for example, to combine single-qubit gates to perform arbitrary rotations in the SU (2) (as described in Sec. 2.6.1). Alternatively, patterns can be combined
via a tensoring operation. This is the case if, for two patterns P1 and P2 , V1 \ V2 = ;.
Definition 8 Two patterns P1 and P2 can be tensored if V1 \ V2 = ;, giving as result a
pattern P12 = P1 ⌦ P2 formed by the concatenation of the commands in P1 and P2 and
computational space given by:
V12 = V1 [ V2 ,
I12 = I1 [ I2 ,
O12 = O1 [ O2 .
(3.4)
The main difference between composing and tensoring two patterns is that in the latter
all unions are disjoint. As a consequence, each command of one pattern commutes with
47
every command in the other one. Throughout this thesis we will be concerned more about
the composition than the tensoring of patterns, since the implementation of complete
algorithms (composed by many gates) stands as a key feature of a universal quantum
computing model, such as the one-way model. For instance, the composition of arbitrary
many measurement patterns implementing the CZ gate and others implementing J(✓)
gates is enough to perform arbitrary quantum computation, since {CZ, J(✓)} is a universal
gate-set for quantum computing.
3.1.3
Command identities
Here I list some identities using commands from the measurement calculus. In the
next section, these identities will be used to re-arrange commands in a given measurement
pattern. Let us start with the identities involving entanglement and correction commands:
Eij Zis = Zis Eij
(3.5)
Eij Xis = Xis Zjs Eij
(3.6)
In both equations there is a qubit i that will be entangled with another qubit j, via
operation Eij . The identities in Eqs. (3.5) and (3.6), which can be easily verified by
multiplying the matrices associated to each command (see Sec. 1.3), say what happens if
the qubits are entangled before the application of the Pauli correction on qubit i. While
nothing changes if the correction is a Zi command (Eq. 3.5), if command Eij switch
places with Xi then a new command Zj appears in the pattern (Eq. 3.6).
The notation t [Mi✓ ]s introduced in Eq. (3.1) can be extended to comprise several
different correction commands:
✓ s r
t [Mi ] Xi
=
✓ s+r
,
t [Mi ]
(3.7)
✓ s r
t [Mi ] Zi
=
✓ s
r+t [Mi ] .
(3.8)
I will also consider the following simplifications in the notation:
✓ s
0 [Mi ]
= [Mi✓ ]s ,
✓ 0
t [Mi ]
= t [Mi✓ ],
✓ 0
0 [Mi ]
= Mi✓ .
(3.9)
Regarding the measurements of Pauli observables, I will consider the following change in
notation:
Mi0 = Mix ,
⇡
2
Mi
= Miy
(3.10)
(3.11)
48
and also the identities:
Mix X s = [Mix ]s = Mix ,
(3.12)
Miy X s = [Miy ]s = s [Miy ] = Miy Z s .
(3.13)
The identity in Eq. (3.12) comes from the simple fact that ( 1)s 0 = 0, while the one in
Eq. (3.13) is a consequence of
⇡
2
=
⇡
2
+ ⇡ (modulo 2⇡). In both Equations I considered
that the measurement angle is given by ( 1)s ✓ + t⇡, where s and t are the X and Z
dependencies, respectively (see Eq. 3.1).
3.1.4
Standardization
In this section I review a technique introduced in [35], known as standardization.
This technique rewrites the command sequence of a measurement pattern, by pushing the
correction commands X and Z to the end (left) and the E commands to the beginning
(right) of the command sequence. The expressions below, obtained by prescribing a
direction (denoted by )) for the application of the identities in Sec. 3.1.3, are known as
rewrite rules and are used in the standardization procedure:
Eij Xis ) Xis Zjs Eij
Eij Zis ) Zis Eij
Eij Xjs ) Xjs Zis Eij
Eij Zjs ) Zjs Eij
(3.14)
(3.15)
(3.16)
(3.17)
✓ s r
t [Mi ] Xi
)
✓ s+r
t [Mi ]
(3.18)
✓ s r
t [Mi ] Zi
)
✓ s
r+t [Mi ]
(3.19)
The following free commutation rules must also be added:
Eij Ak ) Ak Eij ,
Ak Xi ) Xi Ak ,
Ak Zi ) Zi Ak ,
where Ak is not an entanglement operator
(3.20)
where Ak is not a correction operator
(3.21)
where Ak is not a correction operator
(3.22)
If a pattern P0 is obtained from another pattern P by the application of a single
rewrite rule [listed in Eqs. (3.14) to (3.22)], we denote the procedure of doing so by
P ) P0 . Moreover, we denote by P )⇤ P0 the procedure of obtaining P0 from another
pattern P by the application of two or more rewrite rules.
49
Definition 9 (Standardization [35]) A pattern P is called standard if there is no P0
such that P ) P0 . The process of obtaining a standard pattern P from another pattern is
called standardization.
A standard pattern P0 can be obtained in a finite number of steps (i.e., the standardization procedure always terminates). Moreover, all standard patterns are in the so-called
N EM C form5 , which means that the commands in the measurement pattern occurs in
the following order: preparation commands N , entanglement commands E, measurements
commands M and correction commands X and/or Z.
A few comments about the standardization procedure are in order. Regarding the
complexity of the procedure, the standard form of a pattern can be obtained in at most
O(N 5 ) steps, where N is the total number of qubits [35]. Now regarding the importance of
the N EM C form. Since all ancilla preparation and entanglement operations come first in
a standard pattern, it is possible to known right from the beginning exactly which graph
(and hence entanglement requirements) is necessary to perform the computation. It gives
therefore a straightforward way to find out a graph ‘tailored’ to perform the computation
in an economical way, in contrast to the use of cluster states, where several qubits need
to be deleted and others “wasted” just to move a logical qubit state from a given cluster
qubit to another (where a given logical gate takes place). Therefore, the standardization
procedure gives task-specific graphs, which are simpler than typical, universal resources
such as cluster states.
To conclude this section, let us analyze a couple of examples.
Example 1: Teleportation
Let us start with an alternative way of teleporting a qubit along a 3-qubit chain. This
example was discussed in Sec. 2.5.3, where explained how a logical qubit state can be
transferred from one cluster qubit to another. The pattern below is the composition of
5
Remember that in a measurement pattern the commands in the right-most of the sequence are the
ones applied first. Therefore, in the N EM C form the N commands are in the right-most part of the
pattern, while the C commands are the ones in the left-most of part of the pattern.
50
two J-gate patterns (Eq. 3.2), with ✓1 = ✓2 = 0.
(3.23)
J23 (0)J12 (0) =
X3s2 M20 E23 X2s1 M10 E12 )
Eq.(3.14)
(3.24)
X3s2 Z3s1 M20 X2s1 M10 E23 E12 )
Eq.(3.12)
(3.25)
(3.26)
X3s2 Z3s1 M20 M10 E123
Example 2: Single-qubit rotation
Here I compose the measurement pattern that implements the J(✓) gate (Eq. 3.2) in
order to implement a general single-qubit rotation (as in Eq. 2.52):
(3.27)
U (↵, , ) = J(0)J( )J( )J(↵)
The standadization procedure goes as follows:
J45 (0)J34 (
)J23 (
E23 X2s1 M1 ↵ E12 )
Eq.(3.14)
(3.29)
X5s4 M40 E45 X4s3 M3 E34 X3s2 M2 X2s1 Z3s1 M1 ↵ E123 )
Eq.(3.18)
(3.30)
E34 X3s2 Z3s1 [M2 ]s1 M1 ↵ E123 )
Eqs.(3.14) and (3.15)
(3.31)
X5s4 M40 E45 X4s3 Z4s2 M3 X3s2 Z3s1 [M2 ]s1 M1 ↵ E123 )
Eqs.(3.18) and (3.19)
(3.32)
]s2 [M2 ]s1 M1 ↵ E1234 )
Eqs.(3.14) and (3.15)
(3.33)
X5s4 M40 X4s3 Z4s2 Z5s3 s1 [M3 ]s2 [M2 ]s1 M1 ↵ E12345 )
Eqs.(3.18) and (3.19)
(3.34)
X5s4 M40 E45 X4s3 M3 E34 X3s2 M2
X5s4 M40 E45 X4s3 M3
X5s4 M40 E45 X4s3 Z4s2
s1 [M3
X5s4 Z5s3 s2 [M40 ]s1 s1 [M3 ]s2 [M2 ]s1 M1 ↵ E12345
3.2
(3.28)
)J12 ( ↵) =
(3.35)
Determinism in 1WQC
To understand how a deterministic computation can be performed in the one-way
model, let us start by considering what happens if we are not allowed to do any corrections or adaptive measurements. Any such sequence of operations will be of the following
simple form: preparation of qubits in the |+i state, entanglement commands Ei,j , fol-
lowed by a sequence of one-qubit measurements. Since we are looking at sequences with
no corrections or dependencies, these one-qubit measurements can even be performed
simultaneously, resulting in a constant-time computation. On the other hand, the map
implemented on the input-output qubits will be probabilistic, instead of the deterministic,
51
unitary map we would like it to be.
Things would be different if we could somehow choose deterministically which outcome of each measurement Mi✓i actually happens. This is, of course, contrary to quantum
theory; the computational power of such post-selected quantum mechanics has been studied by Aaronson [5], and is believed to be stronger than usual quantum theory. This
notwithstanding, with postselection we can implement the map that projects each qubit
onto the |+✓i i state associated with outcome si = 0:
|
out i
=
Y
|+✓i i
in i
(3.36)
|+✓i ih+✓i |
(3.37)
Pi
i2OC
|+✓i i
Pi
⌘
|
|+✓i i
Note that the map is from input to output qubits, and that the projectors Pi
act
only on a subset of the total number of qubits, projecting the remaining qubits onto the
output state |
out i.
In post-selected quantum mechanics this map could be implemented
deterministically; in the usual quantum mechanics the map can still be implemented, but
with vanishingly small probability, associated with the "right" measurement outcomes
|+✓i i of all measured qubits. The key idea behind the one-way model is to introduce
corrections and adaptive measurements that enable one to implement the map in Eq.
(3.36) deterministically in some cases.
As explained in Sec. 2.4.1, a state’s stabilizer is an operator that leaves the state
invariant. This stabilizer’s property will play an important role since it will allow some
manipulations in the measurement pattern capable of, in some special cases, turn postselected measurements into deterministic, physically implementable patterns. The stabilizer property can be mathematically expressed as:
Ki |Gi = |Gi,
(3.38)
where i labels a vertex from graph G and {Ki } are the (graph-dependent) stabilizers. We
refer to this equation as the application of a stabilizer operator on |Gi.
To find out how to represent a post-selected measurement using measurement calculus,
we will make some map-preserving changes in the command sequence implementing the
unitary J(✓) (Eq. 3.2). We start by observing that in Eq. (3.2), if we remove the
correction X2s1 the implementation of the unitary map would become probabilistic, i.e.,
the map would be correctly implemented only in the cases when the measurement collapses
52
the qubit 1 onto its positive eigenstate |+✓ i. However, we can still get the same unitary
map implementation of Eq. (3.2) if we post-select only the cases when the measurement
result was to collapse onto the "right" eigenvector |+✓ i. Hence, in this context, we can say
that post-selecting a measurement gives the same computation output that one would get
if measure and correct the output in accordance with the measurement’s result, as in Eq.
(3.2). With this relation established, let us manipulate the measurement pattern to get
an interesting representation of post-selection measurements. First, note that the state
to be measured |Gi = E12 N2 |
is stabilized by the two operators {Z11 X21 , Z10 X20 = I},
in i1
as Z1 X2 |Gi = |Gi. In other words, Z1s1 X2s1 is a stabilizer of |Gi independently of whether
s1 = 0 or 1. Then, we can perform the following map-preserving changes on Eq. (3.2):
X2s1 M1✓ |Gi = X2s1 M1✓ X2s1 Z1s1 |Gi = M1✓ Z1s1 |Gi
(3.39)
Since all manipulations done did not change the unitary implemented in the measurement pattern, the last term in Eq. (3.39) is still implementing the unitary J( ✓) between
input qubit 1 and output qubit 2. Note that the last term in Eq. (3.39) has a physically
implausible property: the correction Z1s1 needs to be performed depending on the result
of an yet-unperformed measurement. On the other hand, we know by the first term in Eq.
(3.39) that the map implemented is |
in i1
!|
out i2
= J( ✓)|
in i,
which is exactly the
same map that would be implemented if qubit 1 were projected onto state |+✓ i. Because
of that, the operator Mi✓i Zisi has been formally used in the literature [33, 24, 81, 80, 66, 36]
to represent the projection of a graph state qubit onto the state |+✓i i. Note, however,
that the equivalence of the Mi✓i Zisi operator with a projection operator holds only for the
remaining qubits in the graph state (that is, qubits j 2 G s.t. j 6= i); since qubits are
discarded after they have been measured in the one-way model, it is not necessary that
this equivalence holds also for the qubit been measured. We will explore the physical
meaning of the Mi✓i Zisi operator in Chapter 7; for now I will assume that the following
identity holds:
|+✓i i
Pi
|Gi = Mi✓i Zisi |Gi
(3.40)
where i 2 V . Eq. (3.40) will play a central role throughout this section. It is straightfor-
ward to extend the measurement calculus notation for a set of post-selected measurements:
Y
i2OC
|+✓i i
Pi
|Gi =
Y
i2OC
Mi✓i Zisi |Gi.
(3.41)
53
Although Eq. (3.41) cannot be implemented physically, there are indeed some cases
where the stabilizer operators associated with the graph |Gi properly cancel out all
anachronical commands of form Zisi . In such cases, a deterministic computation can
be physically performed.
To find out if a graph has the required stabilizer operators for deterministic computation, some sets of conditions over a graph were proposed [33, 24, 80, 81]. Those sets of
conditions, known generically as flows, are the main subject of Section 3.3.
The r.h.s. of Eq. (3.40) is clearly an unphysical procedure in the sense that it is
impossible to make the correction Zisi as it depends of the result si of a measurement that
has not been performed yet. To deal with this problem, we will use stabilizer operators
and the fact that the square of a Pauli operator gives the identity. As an example, let us
consider an entangled system represented by a graph G with associated stabilizer operators
Ki . In order to perform deterministic quantum computation we want the measurements
to always collapse onto the same eigenstate. For an arbitrary graph state |Gi, we have:
h+✓ |i |Gi = Mi✓ Zisi |Gi
(3.42)
Now, applying a stabilizer operator on a vertex linked to i by an edge (Kj = Xj
and using Eq. (3.38), we solve the time ordering problem:
Mi✓ Zisi Kjsi |Gi = Mi✓ Zisi Zisi Xjsi
=
Y
k⇠j, k6=i
Y
k⇠j, k6=i
Zksi |Gi
Zksi Xjsi Mi✓ |Gi
Q
k⇠j
Zk )
(3.43)
(3.44)
which is a perfectly implementable procedure, i.e., a runnable pattern (Def. 5).
Since the stabilizer operators are directly related to the graph describing the state’s
entanglement structure, not every command sequence can become time-respecting. Indeed, given a graph state plus the prescription of what measurements we would like to
implement deterministically, it is a well-defined problem to verify if the available stabilizer operators associated to the graph G can make all measurements physically feasible
(as discussed in Sec. 3.3). In other words, we need to cancel out the anachronical Zisi
of Eq. (3.41) for each measurement without introducing new ones. Since each stabilizer
operator is a set of Pauli operators depending on a measurement result, it will induce a
partial time-ordering in the measurement sequence. A partial order reflects the fact that
not every pair of elements in the set must be related. In that way, for some pairs, it may
54
be that neither element precedes the order in the given set. Therefore, in our context, we
can say that a partial time-ordering is the discrimination of which set of measurements
can be done respecting the causality, i.e., allowing only measurements that depend only
on the measurements already performed.
3.2.1
Types of determinism
Due to the probabilistic nature of quantum measurements, not every measurement
pattern implements a deterministic computation – a completely positive, trace-preserving
(cptp) map that sends pure states to pure states. Given a graph G with computational
space (V, I, O), let HV , HI and HO be de Hibert spaces associated with the set of qubits
in V , I and O, respectively. If n is the number of measured qubits (i.e., |OC | = n), then
the run of the associated measurement pattern P may follow 2n different branches. Each
branch is associated to a binary string s of length n representing the collection of classical
outcomes of the measurements on that branch. Moreover, we can associate to each branch
a operator As , called branch map operators, representing the linear transformation form
HI to HO on that branch. Branch map operators can be decomposed in the following
way:
As = Cs ⇧s U
(3.45)
where Cs is a unitary operator collecting all corrections on outputs (that is, a collection
of Pauli operators acting over HO ), ⇧s is a projection from HV to HO representing
the particular set of measurements performed along that branch and Us is the unitary
embedding from HI to HV (preparation and entangling operators). Note that U is the
same for every branch, since the associated graph G is always the same. Therefore, each
measurement pattern realizes a cptp-map:
⌃s A†s As = ⌃s U † ⇧†s Cs† Cs ⇧s U
In other words, T (⇢) =
P
s
(3.46)
= ⌃s U † ⇧†s ⇧s U
(3.47)
= U † ⌃s ⇧†s ⇧s U
(3.48)
= U † U = I.
(3.49)
As ⇢A†s is a completely positive, trace-preserving map (written
as a Kraus decomposition).
A measurement pattern is said to be deterministic if it realizes a cptp-map that sends
pure states to pure states. This is equivalent to saying that branch maps are proportional.
55
In the case where all branch maps are equal, that is, if for every two binary strings si
and sj (defined as above) it holds that Asi = Asj , then we say the pattern is strongly
deterministic. We say a pattern is uniformly deterministic if it is deterministic independently of the values of its measurement angles. Moreover, if a measurement pattern is
deterministic after performing each single measurement and all the corrections depending
on the result of that measurement, we say that pattern is stepwise deterministic. For
simplicity, throughout this thesis I will call patterns which are uniformly, strongly and
stepwise deterministic simply as deterministic patterns.
3.3
Flow theorems
In this section I address the following questions: given an open graph state, can it be
used as resource state for 1WQC? If so, what can be computed using the open graph state
given? To answer those questions, I will review three different sets of conditions which
a graph must satisfy to be used as resource state. In this thesis, those sets of conditions
will be jointly regarded as flow conditions, or flow theorems.
But what does it mean to have a graph satisfying a flow condition? In Sec. 3.2,
I explained how a computation can be encoded as the (unphysical) projections of all
non-output qubit. This idea is encapsulated in Eq. (3.41), which I now reproduce:
Y
i2OC
Mi✓i Zisi |Gi
(3.50)
A set of flow conditions verifies if the geometry of the graph is such that there exists a
set of stabilizer operators able to remove all anachronical Zisi , for i 2 OC , turning the
measurement pattern into a runnable one (Def. 5). If a graph has this property, we say
there is a correction strategy for that graph, which means that there exists a collection of
correction Pauli operators (associated to the stabilizers K of the graph) which enable the
graph to be used as resource state for deterministic 1WQC.
In the rest of this section, I review three different set of conditions, namely regular
flow, generalized flow and maximally-delayed generalized flow. The first one is a sufficient
condition for determinism using open graph states. Moreover, it is also unique: a given
open graph has at most one regular flow. The generalized flow, or simply gflow, is a
necessary and sufficient condition for determinism; it is not unique and has the regular
flow as a sub-case. The last one, the maximally delayed gflow, is sometimes regarded as
optimal gflow because it is the gflow that provides a correcting strategy with the smallest
56
depth. Finally, in Section 3.3.6 I give some concrete examples of the different types of
flows reviewed in this section.
3.3.1
Motivation
Before going through the definitions of different types of flow, I would like to explain
the relevance of the flow theorems. Perhaps the most natural question would be about
the importance of such sets of conditions for determinism, since we already know several
universal graph states. My answer is twofold. First, flow conditions allow one to check
if a given graph state can be used to implement a deterministic computation, that is, to
simulate a unitary evolution. It is specially important in the current technological status
where graph states with more than a dozen qubits are considerably hard to construct.
In this context, the flow conditions allow one to characterize which computations can be
implemented with such restricted graph states, built given a lab’s technological limitations.
The second part of my answer is about the relevance of having several flow theorems. Due to general experimental limitations, it is important to get the most out of the
physical system that a lab is able to produce. And here is where the study of different
sets of flow conditions come into play. Each flow theorem has its own advantages and
disadvantages, that must be evaluated to better fit the lab’s technological limitations.
For instance, regular flow has only one element in the correcting set of each qubit, which
means a reduced classical side-processing during the computation6 . On the other hand,
the maximally-delayed gflow gives the best correction strategy (which in general requires
a more demanding classical side-processing) with respect to the depth of the pattern. It
is, therefore, the depth-optimal flow strategy. Last but not least, the fact that the gflow
function is not always unique might be useful in some scenarios. One example of is a scenario where it is not desirable that a given qubit’s measurement depend on the outcome
of a subset of other qubits in the graph. In a case like this, the existence of a gflow that
respects such a specific requirement would be crucial for the feasibility of the protocol.
3.3.2
Regular flow
The first set of conditions we analyze is called regular flow, or regflow for short. Two
properties of regflow are worth noting before we go to the formal definition. The correction
6
It may be difficult to imagine a scenario where the reduction in the classical processing would be
a relevant feature. Nevertheless, it might result in less-demanding electronic apparatus in some very
specific tasks.
57
1
2
3
1
4
2
3
(a)
4
5
a1
b1
a2
b2
a3
b3
(b)
(c)
Figure 10: Examples of graphs satisfying different flow conditions. In Fig. (a) a graph
with regflow is depicted, in (b) a graph with gflow but no regflow and in (c) a graph with
three different gflows: a regflow, a generalized flow and an optimal gflow (see Section 3.3.6
for more information about this example). Each vertex represents a qubit, where the black
vertices represent measured qubits and the white ones are unmeasured qubits. The input
qubits are represented by boxed vertices and the arrows denote the dependency structure,
pointing from vertex i to its correcting set. The dashed line with an arrow represent the
case where the vertex from the correcting set is a non-neighbouring vertex (vertices not
linked by an edge).
strategy derived from flow’s set of conditions is such that, for each measured qubit i 2 OC ,
just one stabilizer operator Kjsi (for some j 2 I C ) is allowed to be used to get rid of the
anachronical Zisi in Eq. 3.41. This is clearly a limitating condition and because of that,
flow is just a sufficient condition for deterministic computation on an open graph state.
Definition 10 (Regular flow [33]) We say that an open graph (G, I, O) has regular
flow iff there exists a map f : OC ! I C and a strict partial order
f
over all vertices in
the graph such that for all i 2 OC
• (F1) i
f
f (i);
• (F2) if j 2 N (f (i)), then j = i or i
f
j, where N (v) is the neighbourhood of v;
• (F3) i 2 N (f (i));
The set f (i) is often referred to as the regflow’s correcting set for qubit i. Efficient
algorithms for finding regflow (if it exists) can be found in [36, 81]. The regflow function
f is a one-to-one function. The proof is trivial, but as this property is extensively used
in this work, the proof is given below:
58
Lemma 1 Let (f,
f)
be a regular flow on an open graph (G, I, O). The function f is
an injective function, i.e. for every i 2 OC , f (i) is unique.
Proof. Let us assume that for some i 2 OC , f (i) is not unique, i.e. there exists j 2 OC
such that i 6= j but f (i) = f (j). Then according to the flow definition:
j 2 N (f (i)) ) i
f
j,
i 2 N (f (j)) ) j
f
i,
and we arrive to a contradiction because i
f
j and j
f
i cannot be true at the same
time. Hence f (i) has to be unique. ⇤
In the case where |I| = |O|, the regflow function induces a circuit-like structure in a
graph, in the sense that for each input qubit i 2 I, there exists n
0 such that f n (i) 2 O,
where f n is the nth iteration of the function f , and the vertex sequence
{i, f (i), f 2 (i), f 3 (i), ..., f n (i)}
(3.51)
can be translated into a single wire in the circuit model. This circuit-like structure is an
interesting feature of the regflow function, and it will be explored in depth in Chapters 4
and 5. An example of a graph with regflow is shown in Figure 10-a.
3.3.3
Generalized flow
Flow provides only a sufficient condition for determinism but one can generalize the
above definition to obtain a condition that is both necessary and sufficient. This generalization allows correcting sets with more than one element. In those cases, we say
that the graph has generalized flow (or simply gflow ). In what follows let Odd(K) =
{k , |NG (k) \ K| = 1 mod 2} to be the set of vertices where each element is connected
with the set K by an odd number of edges.
Definition 11 (Generalized flow [24]) We say (G, I, O) has generalised flow if there
C
exists a map g : OC ! P I (the set of all subsets of non-input qubits) and a partial order
g
over all vertices in the graph such that for all i 2 OC ,
• (G1) if j 2 g(i) then i
g
j;
• (G2) if j 2 Odd(g(i)) then j = i or i
• (G3) i 2 Odd(g(i));
g
j;
59
The set g(i) is often referred to as the gflow’s correcting set for qubit i. It is important
to note that flow is a special case of gflow, where g(i) contains only one element. This is
a key difference regarding the translation of measurement patterns to quantum circuits,
as I comment in detail in Chapter 4. An example of a graph with gflow (but no regflow)
is shown in Figure 10-b.
The gflow partial order leads to an arrangement of the vertices into layers, where
within a given layer all the corresponding measurements can be performed simultaneously.
The number of layers corresponds to the number of parallel steps in which a computation
could be finished, known as the depth of the pattern:
Definition 12 (Depth of a gflow [81]) For a given open graph (G, I, O) and a gflow
(g,
g)
of (G, I, O), let
Vk
g
8
<max (V (G))
g
=
:max (V (G) \ ([
g
where max g (X) = {u 2 X s.t. 8v 2 X, ¬(u
X according to
(Vk )k=0...d
3.3.4
g
g.
The depth d
g
if k = 0
i<k Vi
g
g
if k > 0
))
v)} is the set of maximal elements of
g
of the gflow is the smallest d such that Vd+1
= ;,
is a partition of V (G) into d
g
+ 1 layers.
Maximally-delayed Generalized flow
In [81] it was shown that a special type of gflow, called a maximally delayed gflow,
has minimal depth.
Definition 13 (Maximally delayed gflow [81]) For a given open graph (G, I, O) and
two given gflows (g,
8k, | [i=0...k Vi g |
gflow (g,
g)
g)
and (g 0 ,
| [i=0...k Vi
g0
g0 )
of (G, I, O), (g,
g)
is more delayed than (g 0 ,
g0 )
if
| and there exists a k such that the inequality is strict. A
is maximally delayed if there exists no gflow of the same graph that is more
delayed.
In this thesis I will simply refer to the maximally delayed gflow as the optimal gflow, given
its property of having the smallest depth possible among the gflows. Note that in [81] it
was proven that the layering of the vertices imposed by an optimal gflow (g,
unique, whereas the gflow itself might not be unique.
g)
is always
60
3.3.5
Constructing a measurement pattern from a flow function
In the last few pages I have reviewed three different types of flow. As explained in
the beginning of the Section 3.3, if a graph satisfies a flow condition it can be used for
deterministic quantum computing. A natural question therefore is how one can construct
a deterministic measurement pattern from a flow. It turns out that a flow function t and
the associated partial order
t
give all the information needed to build the measurement
pattern associated to the open graph. A measurement pattern can be constructed in only
three steps:
1. Write down Eq. (3.41) using the labels for inputs and outputs of the open graph
under consideration;
2. For all i 2 OC , add to the measurement pattern in step 1 the set of stabilizer
operators {Kjsi } (8j 2 t(i)):
Y h
i2OC
Mi✓i Zisi
Y
j2t(i)
i
Kjsi |Gi
(3.52)
3. Remove from the measurement pattern all duplicate Pauli Z and then re-organize
the measurement operators M according to the partial order
t,
moving all Pauli
operators depending on si past the Mi operator.
The final form of the measurement pattern may vary depending on the complexity of the
correcting strategy. In the next section I give some examples of measurement patterns
obtained from different flow functions.
3.3.6
Examples
Consider the open graph depicted in Fig. 10-c. The six-qubit graph has input set
defined by I = {a1 , a2 , a3 } and output set defined by O = {b1 , b2 , b3 }. The flow (f,
f)
of
that open graph is given by:
f (ai ) = bi ,
with partial order a1
f
a2
f
a3
f
(3.53)
{b1 , b2 , b3 }. The depth is therefore df = 3 (disconsid-
ering the output layer). Following the prescription given in Section 3.3.5, the measurement
61
pattern associated to this flow can be constructed:
s
s
s
s
s
Xb3a3 Ma✓33 Za3a2 Xb2a2 Ma✓22 Za2a1 Xb1a1 Ma✓11 |Gi.
(3.54)
The same graph has several different gflows. For instance, a gflow that is different
from both flow and optimal gflow is the following:
g(a1 ) = {b1 , b2 }
g(a2 ) = b2
g(a3 ) = b3
with partial order {a1 , a2 }
a3
g
g
measurement pattern is:
s
s
{b1 , b2 , b3 } and depth dg = 2. The associated
s
Xb3a3 Ma✓33 Za3a2 Xb2a1
+sa2
s
Ma✓22 Xb1a1 Ma✓11 |Gi.
(3.55)
and now there is no dependency between the measurements in qubits a1 and a2 , resulting
in the depth reduction from df = 3 to dg = 2.
Finally, the optimal gflow of the open graph under consideration is given by the
function and partial order (g 0 ,
g 0 ):
g 0 (a1 ) = {b1 , b2 , b3 }
g 0 (a2 ) = {b2 , b3 }
g 0 (a3 ) = b3
with partial order {a1 , a2 , a3 }
is:
s
Xb3a1
g0
{b1 , b2 , b3 } and depth dg0 = 1. The measurement pattern
+sa2 +sa3
s
Ma✓33 Xb2a1
+sa2
s
Ma✓22 Xb1a1 Ma✓11 |Gi.
(3.56)
where no measurement depends on the result of another measurement, yielding the flat
depth dg0 = 1.
62
4
Translating one-way patterns into
the circuit model
It is well-known that the one-way model and the circuit model are equivalent, in the
sense that any algorithm implemented in one model can be efficiently simulated by the
other. However, the way information is processed in each model is radically different:
while the later relies on unitary evolution to drive the computation, the former does so by
performing measurements on a highly entangled state. With such a fundamental difference
between the two models, it is reasonable (and correct) to assume that the translation of
a given computation from one model to the other is not a straightforward procedure in
general.
The meaning of the word translation can be fairly wide in some cases. In almost any
case, one is specially interested in keeping the meaning of what is being translated. It is
so because languages are used for communication purposes, and therefore it’s reasonable
to assume that the meaning of a given text should be the same in both source and target
languages. In the context of computational models, the analogous property, the one which
is desired to be invariant under translation, is the computation itself: if a computation
is able to solve a set of problems in a given model, it is expected that the same set of
problems can be solved when the computation is translated to a different model.
On the other hand, sometimes it is interesting to carry to the target language more
than just the meaning of a text in the source language. In a poem, for instance, it is
usually necessary to preserve structural properties such as metric and rhymes in order
to keep its artistic value. However, if one just wants to pass on the general message of
the poem, the translation of those properties would be irrelevant. Similarly, a translation
procedure for computer programs may or may not preserve some of its properties. For
instance, one could require that, at each computational step, both models have computed
exactly the same (in opposed to just the final result being the same). Therefore, it’s clear
that having translation procedures able to preserve different properties can just enrich
63
the understanding on what are the resources needed to perform a given computation.
In this chapter I review two different translation procedures and explore their limitations and advantages. The chapter is organized as follows. In Section 4.1 I review and
extend the definition of extended circuits, which are easily obtainable translations of measurement patterns. As we shall see, this is a straightforward method but it is inefficient
in some important ways. Next, in Section 4.2, I will describe the so-called star pattern
translation (or simply SP T ). The SP T is a graphic-based translation framework that
gives circuits which are less demanding (in terms of number of qubits) than the corresponding extended circuits. The SP T is, however, a very limited translation framework
since it works only for measurement patterns obtained from a regular flow (Sec. 3.3.2).
In Sec. 4.2.2, I show how the SP T can be used “backwards” to translate circuits to the
1WQC model, resulting in measurement patterns with regular flow. In Sec. 4.3, I explain
exactly why and in which cases the SP T fails. The insights contained in that section
were the starting point of the development of a new translation framework (introduced in
Chapters 5 and with an application given in Chapter 6) and the study of the simulation
of closed timelike curves using the 1WQC model (Chapter 7). I conclude this chapter
with a brief introduction to the new translation framework, which will be introduced and
explored in the next two chapters.
4.1
Extended translation
A straightforward translation method for measurement patterns, which I will refer
to as extended translation, was introduced in [23]. This translation is inefficient, in the
sense that it gives a circuit with as many wires as vertices in the original pattern (instead of having as many wires as input vertices). However its importance comes from
its easy implementation, since the procedure to obtain an extended circuit is simply a
re-interpretation of the measurement pattern commands using circuit notation.
Definition 14 Given a measurement pattern with computational space (V, I, O) and underlying geometry (G, I, O) with a flow (t,
t ).
The corresponding extended circuit C
with |I| input qubits and |V \ I| ancilla qubits, is constructed in the following steps:
1. Each vertex on the graph is translated as a circuit wire;
2. The wires corresponding to I C are prepared in the |+i state;
64
Figure 11: Generic structure of extended circuits obtained from graphs with flow or gflow.
See main text for information about the division into time slices.
3. Each edge linking vertices i and j on the graph (command Eij ) is translated as a
CZij gate;
4. Each measurement Mi✓i is translated as a gate J( ✓i ) in wire i;
5. Each correction command Cjsi (C = Pauli -X or -Z) is translated as Controlled-C
gates with qubit i as the control and j as the target wires. The layering respects
t
by putting the control right after the J-gate and all the corresponding CX acting on
qubit i;
6. All the qubits in OC are measured in the computational basis in the end of the circuit.
Let us divide the extended circuit into time slices and label these time slices as En , Jn ,
Cn , which respectively include the nth round of entangling, J gates and correcting gates,
as illustrated in Fig. 11. We also divide slices Cn into sn smaller slices cn,i , where sn is
the total number of J-gates in slice Jn . Each slice cn,i contains all correction gates with
control on qubit i s.t. Ji is in Jn .
It is easy to verify that the above circuit implements the same computation as the
measurement pattern (see also [23]). For clarity, in what follows we will refer to a CZij
created in Step 3 of Def. 14 as Eij while keeping the notation CZij for those created in
Step 5. Note that by construction, if a pattern is in standard form (Def. 9) then all gates
associated to operators Eij are initially in slice E1 (Step 3 in Def. 14), with E2 , ..., En all
empty.
65
At this point it is important to note some general properties of extended circuits.
They contain gates of only three types: J(✓), CZ and CX. An extended circuit has as
many wires as there are qubits in the associated graph, even though the goal is to implement a unitary on a much smaller subset of qubits. Each non-output qubit undergoes
a single J-gate associated with its measurement in the 1WQC procedure (and wires representing output qubits have no J-gates). The state after each J-gate acts as control of
possibly several CX and CZ gates acting on other wires at a time that is after their initial
entangling gates (step 3) and before their respective J-gates (step 4).
As an example, let us analyze the following pattern, discussed earlier in Sec. 2.5.11 :
X2s1 M1 ✓ E12 | i1 |+i2 . In words, qubit 1 is an input in state | i, qubit 2 an ancilla in
state |+i; we entangle them with a E gate, then measure qubit 1 on basis {|±✓ i ⌘
p1 (|0i±e i✓ |1i)}.
2
Then we correct the state of qubit 2 by applying a Pauli X conditionally
on the measurement outcome s1 . The extended circuit of this measurement pattern
is depicted in Fig. 12-a. Note the coherent Pauli correction: instead of a classically
controlled X gate, we have applied a CX gate controlled by the first qubit’s state. It
is easy to check that the input-output map implemented by the circuits in Figures 12-a
and 12-b is the same; we will call the identity between these circuits the J-gate identity.
This identity will constitute an important building block of the translation framework
developed in Chapters 5 and 6.
Figure 12: Extended circuit for a simple one-way quantum computation protocol. This
J-gate identity will be used repeatedly to simplify generic extended circuits in Chapters
5 and 6. Note that the J gate angles in (a) and (b) differ by a minus sign.
The translation into an extended circuit can be obtained for any measurement pattern,
as the extended circuit is just an interpretation using circuit notation of the 1WQC
operations, with no attempt at optimization or adaptation. It is important to note,
however, that some interesting properties of the one-way model are lost when representing
it as an extended circuit. For instance, all operations in slices C (correction operations) of
an extended circuit are performed classically on the one-way model, while in the circuit
1
See also Eq. (3.2) and associated text.
66
model every operation is a coherent quantum operation.
4.2
Star pattern translation
In this section I review a more spatially economical translation procedure, the socalled Star Pattern Translation (SPT). My approach will be to highlight the circuit-like
structure induced by the regular flow conditions over a graph (Section 3.3.2), showing
how we can use the notion of vertex sequences (Eq. 3.51) to understand how the SP T
works.
4.2.1
From measurement patterns to circuits
Before reviewing the SP T itself, let us first review the concept of vertex sequences in
graphs with regular flow, briefly mentioned in Sec. 3.3.2. Remember that the regular flow
function f induces a partial ordering in the measurement pattern: for every non-output
vertex i there is a vertex f (i) such that i
but different from i, obeys the relation i
f
f
f (i) and every vertex j neighbor of f (i),
j. Hence, the regular flow function defines
‘paths’ in the graph from a vertices tk 2 I to vertices sk 2 O, such that each path can be
represented as a vertex sequence like
{tk , f (tk ), f 2 (tk ), ..., f n (tk ) = sk }
(4.1)
For instance, consider the graph in Fig. 13-a, which has |I| = |O| = 3. From the regular
flow definition (Def. 10), we can come up with three vertex sequences like Eq. (4.1), one
for each input qubit:
{t1 , f (t1 ), f 2 (t1 ) = s1 },
{t2 , f (t2 ) = s2 },
{t3 , f (t3 ), f 2 (t3 ), f 3 (t3 ) = s3 },
(4.2)
(4.3)
(4.4)
These vertex sequences allow us to re-draw the graph in Fig. 13-a so as to follow the
partial ordering: we arrange the vertices from the left to the right respecting the ordering
of the elements in the sets in Eqs. (4.2)-(4.4) and the entanglement structure (edges), as
depicted in Fig. 13-b.
We can get some intuition on how the SP T works by stating that: (1) Since i precedes
f (i) in the partial ordering and they are in the same vertex sequence, it must be translated
67
t1
s1
s3
t1
s1
t2
s2
t3
s3
t2
t3
s2
(b)
(a)
Figure 13: Two isomorphic graphs: (a) generically arranged graph and (b) same graph
re-arranged to obey, from left to right, the partial ordering induced by regular flow.
1
J
J
2
3
J
J
J
J
Figure 14: Quantum circuit obtained from the graph in Fig. 13-b using the Star Pattern
translation.
to the circuit model as an one-qubit gate, which is how we can infer the partial ordering
(the state before a one-qubit gate necessarily precedes the state after it) and (2) edges
between vertices from different vertex sequences represent the two-qubit gate CZ in the
circuit model. With these associations between parts of a graph and circuit elements, we
obtain the quantum circuit in Fig. 14. The angle of each J-gate (omitted in the figure)
is the same as the corresponding measured qubit in the measurement pattern, but with
a minus sign. The circuit in Fig. 14 implements exactly the same computation as the
one-way computation using the graph state in Fig. 13-a or -b.
More generally, we can decompose any graph with regular flow in several subgraphs,
with one subgraph per measured qubit (Fig. 15-a). Those subgraphs can be directly
translated into the circuit model, resulting in the circuit in Fig. 15-b. It is easy to verify
that the one-way computation performed using the graph in Fig 15-b is the same as the
one using the circuit in Fig. 15-b. These special subgraphs are associated with simple
command sequences of the format:
X2s1 M1✓1 E12 E13 · · · E1n ,
(4.5)
where vertex 1 is a measured qubit and 2, · · · , n are neighbors of qubit 1 in the graph,
68
2
J
1
3
4
1
n
2
3
4
n
(a)
(b)
Figure 15: Fig. (a) shows a subgraph that works as a building block for the Star Pattern
translation. Decomposing a general graph into these subgraphs, we can translate each one
to the corresponding circuit (Fig. b) and compose them to obtain the complete translated
quantum circuit.
as in Fig. 15-a. Thus, when we have a measurement pattern associated to a graph with
regular flow, this means that the pattern can be fully decomposed into sub-patterns of
the form of Eq. (4.5). Once we identify these pattern building blocks, the translation to
the circuit model is fairly simple. The composition of the circuits originated from those
subgraphs will give a quantum circuit that performs the same computation as the graph
from which it was obtained. This procedure can be formally defined as follows [33]:
Definition 15 (Star Pattern Translation [33]) Let G be an open graph with computational space (V, I, O) and regular flow (f,
f ).
The corresponding quantum circuit CG
can be obtained by following the steps below:
1. Pick a vertex i in the first level of the partial order. Associate to vertex i a subgraph
Si constructed by taking vertex i as the input and all its neighbor vertices in G as
output. Delete vertex i from G and repeat the process for all vertices i 2 OC , always
respecting the partial order;
2. Translate each subgraph generated in Step 1 to the corresponding quantum circuit
(as in Fig. 15), respecting the labels;
3. Combine all circuits obtained in Step 2 by connecting segments of wires with the
same label.
It is important to note, however, that the SP T cannot be used when the correcting
set has more than one element, since we are not able to uniquely identify paths like Eq.
(4.1) anymore. Also, from the measurement pattern framework, it means that we can not
69
rearrange the pattern into the simple building blocks like Eq. (4.5). Therefore, we can
not apply the SP T if we want to obtain a circuit from a graph with gflow (but no regular
flow). In Sec. 4.3 I show some results on the understanding of this limitation. Before
analyzing the problem itself, let us see how the concept of star patterns can be used also
to translate computations from the circuit model to the one-way model.
4.2.2
From circuits to measurement patterns
The decomposition of a measurement pattern into star patterns can also be used to
translate quantum circuits to the one-way model. Due to the limitations on the applicability of the SP T (discussed in Sec. 4.2.1), the translation of quantum circuits using this
method results always in patterns associated to graphs with regular flow. The translation
from circuits to measurement patterns is implemented in a graphical level, following the
steps in the definition below.
Definition 16 (Definition 5.2 from [23]) Let C be a circuit on n logical qubits composed by gates from the universal gateset {J(✓), CZ}. The corresponding open graph state
and associated measurement pattern P can be obtained by the following steps. Starting
from the left of the circuit we do what follows:
1. Replace each J-gate by a horizontal edge connecting two vertices. Draw an arrow
on the edge and label the vertex on the left as “input” and the one in the right as
“output”;
2. Replace each CZ gate by a vertical edge connecting two vertices. Label both vertices
as “input/output”;
3. The components generated during steps 1 and 2 must now be connected to form
the final graph. Starting from the left, contract two non-adjacent vertices at a time
according to the subcases listed below:
• Two vertices labeled input/output are contracted as one vertex with input/output
label;
• A vertex labeled “input/output” and a vertex labeled “input” are contracted as
one vertex with “input” label;
• A vertex labeled “output” and a vertex labeled “input/output” are contracted as
one vertex with “output” label;
70
{
J(a)
Input
J(b)
In
In/Out
Steps 1
and 2
Out
In/Out
In
}
Output
In/Out
In/Out
In/Out
J(c)
In
Out
In
Out
In/Out
Out
a
In
Aux
b
c
Out
Step 3
In/Out
Figure 16: Using the method in Def. 16 to generate an open graph state able to simulate
the circuit on the top of the figure.
• Two vertices labeled “output” and “input” are contracted as one vertex with the
label “auxiliary”.
In Fig. 16 we depict the application of the method described in Def. 16. Starting
with the circuit in Fig. 16-a, Steps 1 and 2 of Def. 16 are applied, resulting in the
set of subgraphs in Fig. 16-b. Then, those subgraphs are connected according to the
instructions in Step 3 of Def. 16, resulting in an open graph state able to simulate the
original quantum circuit. The associated measurement pattern can be inferred directly
from the resulting open graph state, since both computational space (V, I, O) and flow
function f are graphically specified. This method can be applied for any quantum circuit
decomposed into the gateset {J(✓), CZ}, yielding always graphs with regular flow.
71
4.3
The problem of using star pattern translation for
patterns with no regular flow
In order to appreciate the problem that arises when applying Star Pattern Translation
for gflow measurement patterns, let us apply it to the gflow pattern associated to the graph
in Fig. 17-a and see what happens. First, remember that this translation method consists
in decomposing a general graph into building blocks like Fig. 15-a and then substituting
it by the equivalent circuit (Fig. 15-b). The composition of these sub-circuits gives the
complete translated circuit. The application of this method for our example is shown
in Fig. 17. There I have decomposed the complete graph into Star Pattern building
blocks, resulting in the subgraphs shown in Figs. 17-b and 17-c. These subgraphs are
equivalent to the circuits in Figs. 17-d and 17-e, respectively, that can be composed
together (respecting the labeling) into the circuit in Fig. 17-f.
As can be noted, the translation of a perfectly runnable measurement pattern resulted
in an anachronical circuit, which puts in question the validity of SP T for gflow patterns.
This unexpected result for gflow pattern has been pointed out in the papers [24] and [66],
where the authors claimed that it may suggest an extraordinary efficiency of gflow patterns, since the circuit originated from SP T has anachronical gates in order to preserve
the same computational map of the associated measurement pattern. In this section I will
show that the SP T simply does not account for every correction in the gflow’s correcting
sets and therefore it does not constitute a complete translation method for gflow patterns.
As shown in Section 3.3.3, gflow patterns have a more complicated dependency structure,
since they allow for several stabilizer operators in the measurement’s correcting set (differently from the regular flow, which has just one). The SP T is ineffective for gflow patterns
due to the impossibility to define a circuit-like structure in the graph (which is naturally
achieved by regular flow function), always resulting in compact circuits with anachronical
gates.
In order to get some insight on the origin of this problem, let us decompose the gflow
measurement pattern associated to the graph in Fig. 10-b into star patterns (like Eq.
4.5). The correcting sets are g(1) = {3} and g(2) = {4, 5} and therefore the measurement
pattern can be written as
X5s2 X4s2 X3s1 M2✓2 Z2s1 M1✓1 E23 E24 E14 E15 E13 .
(4.6)
We can conveniently rewrite this measurement pattern using the rules from Eqs. (3.5)
72
5
5
1
5
J
3
3
1
4
(d)
4
(b)
5
5
3
1
3
J
1
J
2
2
(a)
4
1
J
3
J
2
4
4
3
(f)
3
2
2
J
(g)
4
(e)
4
(c)
Figure 17: Graphical representation of the star pattern translation being applied to the
gflow pattern associated to the graph in (a). Figures (b) and (c) show the star pattern
decomposition for the graph in (a). These subgraphs can be directly translated to the
circuit model using the correspondence shown in Fig. 15. As a result we have the circuits
in (d) and (e). Finally, composing these subcircuits we obtain the quantum circuit in
(f). We can rewrite this circuit, preserving its meaning, to the circuit in (g) by adding
a SW AP gate acting on the (newly added) time-traveling qubit [the round wire at the
bottom of Fig. (g)] and the left-most wire segment where the anachornical CZ in Fig. (f)
were acting upon. By doing so, the quantum state just before the SW AP gate is able to
go to the future, be acted upon by the CZ gate and then go back to the past and re-enter
the time respecting part of the circuit using the same SW AP gate. The circuit in (g) has
no anachronical CZ like the circuit in (f) but instead there is a quantum wire interacting
with a closed wire (which represents a qubit traveling in time).
and (3.6) in order to highlight the star patterns (Eq. 4.5):
X5s2
|{z}
Lost inf ormation
X4s2 M2✓2 E23 E24 X3s1 M1✓1 E14 E15 E12
|
{z
}|
{z
}
Star pattern 2
(4.7)
Star pattern 1
As can be noted in Eq. (4.7), the star pattern decomposition will only account for one
stabilizer operator per measurement, which in this case are the ones associated to the
vertices s(1) = {3} and s(2) = {4}. The information that would be lost in translation
is the stabilizer operator K5s2 = X5s2 Z1s2 . However, in Eq. (4.7) only the correction X5s2
is seen because Z1s2 has been canceled out with the Z1s2 from the stabilizer K4s2 , since
Z1s2 Z1s2 = 1.
Due to the construction of SP T , the lost information will always consist of stabilizer
s
operators {Ki j } which, as we know, do not change the map implemented by the compu-
73
s
tation, since Ki j |Gi = |Gi for i 2 I C . However, some of those stabilizer operators are
needed to eliminate anachronical Pauli corrections. Consider, for example, the graph in
Fig. 10-b. As we have already seen, the SP T is not able to translate the corrections
due to the stabilizer operator K5s2 = X5s2 Z1s2 , which means that it translates, in fact, the
following measurement pattern:
X4s2 X3s1 M2✓2 Z2s1 M1✓1 Z1s2 E23 E24 E14 E15 E12
(4.8)
where the last equation is obtained by removing the stabilizer operator K5s2 from the
measurement pattern in Eq. (4.7) (or re-applying it, since K5s2 K5s2 = 1). Note that now
the measurement pattern has the Pauli operator Z1s2 , which is an operation on qubit 1
that depends on the result of a measurement (on qubit 2) not yet performed. In other
words, the measurement pattern has become anachronical, not physically implementable.
Remember that the equivalent quantum circuit, obtained applying the SP T directly to
the graph in Fig. 17-a is shown in Fig. 17-f. Note that the anachronical Pauli operator
Z1s2 of Eq. (4.8) appears as a controlled-Z acting at different times in Fig. 17-f.
It is interesting to note that the anachronical circuit in Fig. 17-f can be manipulated to give an equivalent circuit without anachronical gates but that interacts with a
qubit traveling in time, as seen in Fig. 17-g. This representation stands for an ordinary
quantum evolution in the presence of a closed time-like curve (CTC), which is a possible
solution to Einstein’s General Relativity equations. Although the quantum circuit in Fig.
17-g cannot be implemented in a lab, the relation between measurement-based quantum computation and quantum circuits with time-traveling qubits can give interesting
insights. This topic will be extensively discussed in Chapter 7. There I show how closed
time-like curves appear naturally in the 1WQC context, studying some CTC circuits using 1WQC techniques and then comparing the results with the two main CTC models:
Deutsch’s [39] and the Bennett-Schumacher-Svetlichny (BSS) model [1, 106]. The oneway model encompasses the latter, having a general disagreement with the well-known
Deutsch model. Furthermore, a characterization of a class of CTC circuits that admit
deterministic simulation is given using stabilizer techniques associated with the one-way
model.
74
4.4
Alternative approaches to the translation problem
Since the star pattern translation does not work properly for measurement patterns
whose underlying graph has no regular flow, a new translation procedure is necessary for
those patterns. An alternative approach to this problem was proposed recently in [43],
where the use of a diagrammatic calculus allows one to rewrite any graph with gflow as
a graph with regular flow, which can then be translated into the circuit model using the
method described in Sec. 4.2.2.
It would be useful, however, to have a translation procedure that does not need an
"encoding" into another formalism (such as the diagrammatic calculus). Such translation would make it easier to understand what is being changed during the translation
procedure, since the circuit model is regarded as more intuitive. In the next chapter, I
develop a new translation framework which does not use the SP T as a sub-routine nor
the diagrammatic calculus, and yet gives the same results for graphs with regular flow,
being also applicable to at least some graphs with gflow. In Chapter 6, I use this new
translation framework to optimize quantum circuits by back-and-forth translation to the
one-way model.
75
5
Compact circuits from one-way
quantum computation
In this chapter I introduce a new translation framework for one-way quantum computation [40, 42]. In the previous chapters, I have reviewed some properties and limitations
of the existing translation procedures: the star pattern translation (Sec. 4.2.1) gives
time-respecting circuits only for a restricted class of 1WQC algorithms, namely the ones
associated to regflow measurement patterns. The translation to extended circuits (Sec.
4.1), on the other hand, is able to translate all 1WQC algorithms but is remarkably inefficient in what concerns spatial resources. Moreover, the method developed in [43] (which
relies on a diagrammatic calculus to do the translation) also works for any 1WQC algorithm, although with no concern in keeping some important algorithm properties, such as
its depth.
The translation framework developed here relies on the relationship of the gates in the
extended circuit’s time slices E with the ones in the correction slices C, as they appear via
the use of the stabilizer formalism. This relationship is used to rewrite extended circuits
with the purpose of revealing the information processing that is taking place at the logical
level, that is, with logical qubits. I introduce two novel algorithms able to remove |OC |
many wires from the extended circuits, while preserving the number of time slices J and the
computation being implemented. Each algorithm is designed to work for a different class
of measurement pattern: The first, designed to deal with the regflow correcting structure,
gives as output circuits equivalent (in terms of number of wires and computation being
implemented) to the ones given by the star pattern translation. The second algorithm
works for a larger class of measurement patterns, namely the ones associates to SignalShifted Flow (SSF)1 . In both cases the structure within each J layer is intentionally kept
unchanged, allowing a better appreciation of how the number of computational steps in
each model compare to each other.
1
Recently introduced by the author and two collaborators in [42].
76
This chapter is divided as follows. In Section 5.1 I introduce the concept of compact
circuits and the method for obtaining them from extended circuits, which I call compactification procedures. Section 5.2 is reserved to the development of the compactification
procedure for regflow extended circuits. First, in Sec. 5.2.1, I review the regflow definition and comment on some important properties of regflow extended circuits. The
algorithm that implements the compactification procedure for regflow is introduced in
Sec. 5.2.2; A comparison with the existing method for regflow translation, namely Star
Pattern Translation (see Sec. 4.2.1), and an example of the algorithm in use are also
provided in this section. In the beginning of Section 5.3, I introduce a new type of flow
called Signal-shifted Flow (SSF) and explore several of its structural properties. The SSF
is more general than the regflow and its limitations and advantages are discussed in the
section. An algorithm to implement the compactification procedure for SSF extended
circuits is given in Sec. 5.3.5. Finally, in Sec. 5.4 I comment on the difficulty of designing
compactification procedures for arbitrary flows and give a couple of examples of gflow
extended circuits being transformed into their compact versions.
5.1
Compactification procedures and circuit rewriting
rules
The goal of a compactification procedure is to “extract” from an extended circuit the
array of gates that is being applied to the logical qubits, without explicitly calculating
what the computation being performed is. In this thesis, I will call this array of gates the
compact circuit.
There is a simple pattern from which the compact circuit can be easily obtained from
its extended circuit. This is done by explicitly calculating the action of every gate in the
extended circuit. This pattern was analyzed twice in this thesis (Secs. 2.5.1 and 4.1) and
is called the J gate identity: X2s1 M1✓1 E12 N2 . The associated extended circuit is in Fig
18-a. As explained before, it implements the J-gate and therefore the circuit using only
logical qubits is the one shown in figure 18-b.
This method of explicitly calculating the action of every gate in the extended circuit
(and also the final round of measurements in the computational basis) in order to find
out the associated compact circuit is way too demanding to be applied to large circuits.
Moreover, once the calculation is done, one would have only the full unitary being applied
to the logical qubits; the decomposition of such unitaries into a universal gate-set - in
77
Figure 18: The J-gate identity. This is the same identity as the one shown in Sec. 4.1,
and it is reproduced here for convenience of the reader.
order to facilitate its implementation in a physical system - is an exponentially hard task.
However, the compact circuit associated to the extended circuit in Fig. 18-a (shown in
Fig. 18-b) has the remarkable property of being itself already decomposed in the universal
gate-set {J(✓), CZ} - the same used in extended circuits. In the following sections, I
develop procedures able to rewrite large extended circuits until |OC | many J-gate identities
can be applied, resulting therefore in a compact circuit. This rewriting process, followed
by the application of J-gate identities, is called compactification procedure.
A remark on notation. In this chapter, I will refer to every CZ gate the in E slices
of an extended circuits as “E gates”. The CZ gates in C slices will be called just “CZ
gates”. Moreover, every extended circuit analyzed in this chapter is assumed to have been
obtained from a pattern in the standard form. As a consequence, every E gate is initially
placed in the first E slice, that is, E1 .
5.1.1
Compactification procedures
Compactification procedures can be described as a way of globally rewriting extended
circuits in order to remove wires from the circuit without changing the computation being
implemented. One way to achieve this is to rewrite the circuit to create J-blocks, which
I now define.
Definition 17 (J-block) Let P be a measurement pattern with underlying geometry (G, I, O)
and corresponding extended circuit C. We say there is a J-block on wires i and j if the
following set of conditions is satisfied (see Fig. 18-a):
1. The initial state of wire j is |+i.
2. The gate sequence (Eij , Ji (✓i ), CXij ) appears in C.
3. The only gate acting on the wire j before CXij is Eij .
78
4. The only gates acting on wire i after Eij are Ji (✓i ) and CXij (in this order).
5. After the CXij gate, qubit i is measured in the Z basis.
Once a J-block is created (via circuit rewriting), one can use the identity in Fig. 18
(J-gate identity) to remove one wire from the circuit. In general, extended circuits are not
prepared for direct applications of the J-gate identity. In Definition 14, all corrections Cjsi
are translated as controlled-gates with control placed after the Ji gate. Hence Condition 4
in Definition 17 is not satisfied in general. Moreover, since all E gates are initially placed
in slice E1 , Condition 3 is not satisfied for general extended circuits either (see example
in Figure 32-b). Thus, it is clear that in order to remove wires from extended circuits, we
first need to rewrite the cicuits.
In order to create J-blocks in extended circuits, we explore the relationship between
the E gates and the correcting gates C, since the latter are defined according to the former
via the stabilizer formalism. In other words, there is a direct relationship between the
gates in slice E1 and all other two-qubit gates in the rest of the extended circuit. In the
case where we succeed in removing as many wires as there are non-output qubits in the
graph, we say that the resulting circuit is in a compact form. We will refer to circuits in
the compact form as compact circuits.
Definition 18 (Compactification procedure) We call compactification procedure the
process of rewriting extended circuits until the number of J-blocks equals the number of
non-output qubits in the associated open graph, followed by the application of the J-gate
identity (Fig. 18) to each J-block.
In the next section I propose a set of circuit identities with the purpose of exploring
the aforementioned relationship between two-qubit gates in extended circuits in order to
create J-blocks.
5.1.2
Circuit rewrite rules
The goal of the circuit identities introduced in this section is to rewrite extended
circuits, allowing the application of J-gate identities and thus the remotion of auxiliary
wires. In general, each application of the J-gate identity requires a few preparatory circuit
manipulations. To see why, recall that corrections associated with a given measurement
Mi appear in the extended circuit as two-qubit gates (more specifically, CXs and CZs)
79
Figure 19: Extended circuit representation of a measurement and the corrections it requires on other qubits for deterministic computation in the 1WQC model. Note that
each correction arises from a particular stabilizer in the correcting set of i, raised to the
power si , where si is the outcome of measurement i. This sub-circuit shows only the gates
corresponding to the correcting set of qubit i.
controlled by the state after the J-gate on qubit i, as in Fig. 19. While in the circuit of
Fig. 18-a there is just a single CX and no CZ correction, in general extended circuits
there may be other CX and CZ gates that will need to be removed if we are to use the
J-gate identity to simplify the circuit.
In Fig. 20 we introduce five circuit identities that aid us in this task. The identities
in Figs. 20-b and 20-c can be easily obtained from the one in Fig. 20-a, as we describe
now. To start, note that the circuit identity in Fig. 20-a can be written as:
CZik CXij CZjk = CZjk CXij
(5.1)
with i, j and k labeling arbitrary qubits (or vertices in a graph). Note that in this
representation one must read the gates from the right-hand side to the left-hand side,
while in the circuit model it is the other way around. If qubit j is in state |+i, and since
CXij |+ij = |+ij , we have:
CZik CXij CZjk |+ij = CZjk CXij |+ij = CZjk |+ij
(5.2)
multiplying on the left by CZik , omitting the |+ij and swapping the sides (to match
Figure 20-b), we have:
CZik CZjk = CXij CZjk
(5.3)
which is exactly the circuit identity of of Fig. 20-b. From Eq. (5.1), considering Hk =
80
1i ⌦ 1j ⌦ Hk and Hk being the Hadamard gate applied to qubit k, we also have:
(5.4)
Hk [CZik (Hk CXij Hk ) CZjk ]Hk = Hk [CZjk ]Hk CXij
Since Hk CZik Hk = CXik , we get:
(5.5)
CXik CXij CXjk = CXjk CXij
which is the circuit identity in Fig. 20-c. I also consider the adjoint of the circuit identities
in Fig. 20-a and 20-c, which are depicted in Fig. 20-d and 20-e, respectively.
i
i
i
j
j
j
k
|+i k
(a)
k
|+i
(b)
(c)
i
i
j
j
k
k
(d)
(e)
Figure 20: Five circuit identities. The circuit identity in Figure (b) is true only if qubit
k is initially in the |+i state. The circuit identity in Figure (d) [Figure (e)] is obtained
by multiplying CZik (CXik ) in both sides of the identity in Figure (a) [Figure (c)].
The circuit identities introduced in this section will be used as part of the compactification algorithms for Regular Flow (Section 5.2) and Signal-shifted Flow (Section 5.3).
We will use the identities by substituting the subcircuit on the right for the one on the left
in extended circuits. The two-qubit gates we need to remove from the extended circuit in
order to apply J-gate identities are part of the correction structure of the original measurement pattern. Since the correction structure differs for graphs with different flows,
each case must be analyzed separately.
5.2
Compact circuits from graphs with regular flow
In this section I present the compactification procedure for regflow extended circuits.
First, in Sec. 5.2.1, I analyze several structural properties of the regflow function and,
consequently, of regflow extended circuits, that will shed light on which rewrite rules shall
81
be used in order to create |OC | many J-blocks in the extended circuit. In Sec. 5.2, a full
compactification algorithm for regflow extended circuit is given. Finally, a comparison
with the star pattern translation (SPT, for short) is given in Sec. 5.2.2.3.
5.2.1
Regular flow structural properties
Let us start by briefly reviewing the definition of regflow. Consider a graph G for
which we define a set I of input vertices and a set O of output vertices. We define a
function f : OC ! I C (from measured to prepared qubits) and a partial order
i
f
f,
where
j means that qubit j must be measured after qubit i. We say graph G has regflow
if for each vertex i 2 OC , we can define f such that:
(F1) i, f (i) 2 G;
(F2) i
f
f (i);
(F3) For each k neighbor of f (i) in G, with k 6= i, we have i
f
k.
When the entanglement graph has regflow, f is called the regflow function and identifies the stabilizer Kfsi(i) that corrects each measurement i, and which will be translated
in the extended circuit as one CX and one or more CZ gates (according to the number
of neighbors of f (i) in G) controlled by qubit i.
Now let us analyze how the properties of the regflow function reflect in the design
of regflow extended circuits. Regflow’s condition (F2) implies that, for any wire w such
that there is a CX target acting upon it in slice Ci , the gate Jw has to be in slice Ji+1
or later in order to respect the partial order
f.
Equivalently, regflow’s condition (F3)
implies the same for CZ in a given slice Ci . In Fig. 21-a, for instance, this means that
the gates Jj and Jk must be placed in slice Ji+1 or later, since there are correcting gates
acting upon wires j and k in slice Ci . Moreover, since regflow has just one element in its
correcting set (|f (i)| = 1), each cn,i slice in the associated extended circuit has just one
CX gate but possibly several CZ gates (In fact, |N [f (i)]|
1 many2 ). In the next section
I develop a compactification procedure based on these properties.
5.2.2
A compactification algorithm for regular flow extended circuits
In this section I propose an algorithm able to creat |OC | many J-blocks in a regflow
extended circuit, allowing the use of J-gate identities for obtaining compact circuits. For
2
where N [f (i)] denotes the set of neighbors of vertex f (i) in G.
82
...
...
J
i
i
...
J
i
j
j
j
k
k
k
(a)
(b)
J
(c)
Figure 21: Rewrite Procedure 1. In Fig. 21-a we show the undesired CZik in time slice
Ci with the corresponding, initial entangling-round CZjk in time slice E1 . In Fig. 21-b
the CZjk gate was moved from E1 to Ci where the circuit identity in Fig. 20-a can be
applied, resulting in the circuit depicted in Fig. 21-c.
graphs satisfying the regflow condition this translation into compact circuits is equivalent
to the star pattern translation proposed in [23], as discussed in Sec. 5.2.2.3.
5.2.2.1
A rewrite procedure for regular flow
Let us start by recalling how the corrections in an extended circuit are associated
with the graph’s geometry. As we discussed in Section 3.2, to correct for the probabilistic
Q
character of the measurement on qubit i we identify operators Kjsi = Xjsi k⇠j Zksi , where
k are the neighboring vertices of a given vertex j in the graph. In graphs with regflow,
a single operator Kjsi is required to correct for the measurement on i, and the vertex j
associated with it is always adjacent to i. As depicted in Fig. 19, the correction operator
Kjsi is translated in the extended circuit as a single CXij gate together with a collection
of CZik gates, one for each k 6= i adjacent to j in the graph. There is also a Zisi operator
in Kjsi , but this operator does not translate as a gate in the extended circuit; instead, it
cancels the anachronical Zisi operator associated with the deterministic projection (Eq.
3.41), as described previously in Sec. 3.2. As the stabilizer operators reflect the graph’s
geometry, for each k 6= i that is adjacent to j in the graph there must be a Ejk gate at
the beginning of the extended circuit, that is, in slice E1 .
This means that for each CZik gate in C we would like to remove (in order to create
J-blocks), the extended circuit is guaranteed to have a previous Ejk (in slice E1 ) and also
a CXij in C (that commutes with the CZik gate). These three gates can be conveniently
transformed using the circuit identity in Fig. 20-a, which swaps gates CXij and Ejk , while
eliminating the troublesome CZik gate. This procedure, of bringing the corresponding E
from the beginning of the extended circuit to alongside the CZ we need to remove, followed
by the application of the circuit identity in Fig. 20-a, is called Rewrite Procedure 1 (or
83
RP 1 for short) and it is depicted in Fig. 21.
Now I would like to show that this removal of unwanted CZ gates can be done to any
regflow extended circuits. In what follows I give a brief description of what is formally
developed in Sec. 5.2.2.2 - the procedure for obtaining compact circuits from regflow
extended circuits. Using the following prescription all unwanted CZs can be removed. In
what follows, let Wi be the set of wires acted upon by the J-gates in slice Ji . First, move
every E gate in E1 not acting on a wire in W1 to slice E2 , commuting with the gates in
C1 . This commutation is either trivial (no gates to be commuted) or can be done using
the identity in Fig. 20-a, as done in Fig. 21. With this, all unwanted CZ gates in C1
are removed. Now, since all E gates from the entanglement structure are in E2 (except
for those acing on wires in W1 ), the same procedure can be repeated to remove the CZ
gates in C2 , and so on until we are done. Note that, in each step, all E gates needed
to remove the undesired CZs of a given slice Ci are placed in slice Ei , which allows the
application of the circuit identity in Fig. 20-a. Repeating this procedure for all slices,
all unwanted CZ gates can be removed. In the next section I give an algorithm that
implements the procedure described above3 and show that it creates |OC | many J-gate
blocks in the extended circuit.
5.2.2.2
The algorithm
Here I give an algorithm that implements the regflow compactification procedure,
described in the last section. First, let me define a layering function for regflow:
Definition 19 Let G = (I, O, V ) be a graph with regular flow function (f,
f ).
Define
Lf (i) to be a layering function from OC into a nonzero natural number:
Lf (i) =1,
if
Lf (i) =n,
if
@j 2 V
max
j2V s.t.j
fi
s.t. j
[Lf (j)] = n
f
i
(5.6)
1
(5.7)
Due to the recursive character of Def. 19, in order to evaluate Lf (i) for all i 2 OC one
must start with vertices in the set {f (i) [ N [f (i)] \ {i}} for i such that Lf (i) = 1 (Eq.
5.6), and then moving forward doing the same for Lf (i) = 2, Lf (i) = 3 and so on. Now
let us see how Algorithm 1 works.
3
The algorithm differs from the description in Sec. 5.2.2.1 in just one point: the algorithm only moves
gates from slice En to En+1 if such procedure involves the application of a circuit identity. It is clear that
doing so has no effect on the resulting compact circuit, but makes Algorithm 1 simpler.
84
Algorithm 1: Transforms a Regular Flow extended circuit into a compact circuit.
Input: Regular Flow extended circuit.
Output: Compact form of the inputted extended circuit.
1 begin
2
D
maxi2OC {Lf (i)}
3
for n = 1 to D do
4
Let Wn = {an,1 , ..., an,mn } be the set of the mn wires with J-gate in slice Jn
5
forall the an,i 2 Wn do
6
forall the k 2 {N (f (an,i )) \ {an,i }} do
7
i
an,i ;
8
Apply RP1;
9
Apply the J-gate identity for all pairs of wires {i, f (i)} such that i 2 OC .
Lemma 2 Algorithm 1 is sound and outputs compact circuits.
Proof: The algorithm analyzes one layer of the extended circuit at a time, from n = 1 to
n = maxi2OC {Lf (i)} (last layer containing non-output qubits). In each layer n, a RP1 is
applied to the triad (i, j, k) where i 2 Wn (the set of wires with J-gate in slice Jn ), j
f (i)
and k 2 N (f (i)) \ {i}. Within each layer, the application of a RP1 to wires (i, j, k) is
independent from the application to another triad (i0 , j 0 , k 0 ) due to the following reasons:
(i) there are no two CX gates acting with target on the same wire (since f is injective)
and therefore every gate in Cn commute to each other; (ii) Since all E gates (that will be
used for the application of a RP1) commute with each other, they can be pushed to slice
Cn in any order.
Now let us analyze what changes in each layer n after the application of all required
RP1’s. For every wire i 2 Wn the following statements hold: (i) the only gate acting
on wire i after the J-gate is CXif (i) (all CZ gates have been removed); (ii) Every gate
acting on wire j = f (i) before CXif (i) were moved forward in the circuit with the only
exception being the gate Eif (i) . In other words, after each iteration of the for loop the
algorithm creates |Wn | many J-blocks in the extended circuit. Since the for loop runs D
P
C
C
times and D
n=1 |Wn | = |O |, when the algorithm exits the for loop it has created |O |
many J-blocks. Finally, in the ninth line of the algorithm |OC | many J-gate identities are
applied, giving a compact circuit as the output of the algorithm.
⇤
85
5.2.2.3
Comparison with star pattern translation
In what concerns the final compact circuit, the compactification procedure for regflow
extended circuits is equivalent to the so-called star pattern translation [33], a graphicalbased approach for translating graphs with regflow. To see this, first remember that
an extended circuit has |V | wires, where |V | is the number of vertices in the associated
graph. We have shown that for any regflow pattern our method removes |OC | wires
(|OC | = number of measured qubits) from the extended circuit, resulting in a simplified
circuit with |V |
|OC | = |O|
|I| wires. Moreover, Algorithm 1 removes all CX and CZ
gates initially in slices C, while keeping all E gates but the ones used for the application
of J gate identities. Hence, the number of J gates plus the number of CZ in the final
compact circuit equals the number of edges in the associated graph. The same is true
for the star pattern translation method. The two procedures give circuits implementing
the same computation with the same number of wires, and therefore both can be used to
obtain compact circuits for regflow measurement patterns. One of the advantages of our
method when compared to SP T is that it can be extended to some graphs that do not
have regflow but which satisfy the gflow conditions, as I discuss in the next sections.
5.3
Compact circuits from graphs with signal-shifted
Flow
In this section I introduce a new type of flow, which I call Signal-shifted Flow (SSF),
and give a compactification procedure for extended circuits obtained from patterns with
this flow. In short, a SSF pattern is a pattern obtained by applying the signal-shifting
rules (Section 5.3.1) to a regular flow pattern. Therefore, only graphs with regular flow
have SSF. In what follows, I briefly review the signal-shifting rules and then define the
SSF (Sec. 5.3.2). Still in the same section, I explore some properties of SSF and comment
on two specially interesting ones: (i) SSF is a generalized flow and (ii) if a graph has
regular flow, the SSF provides the (time) optimal correcting strategy. In Section 5.3.3,
I define a few structural properties of SSF that will be useful in the construction of a
compactification procedure for SSF extended circuits, which is the subject of Sections
5.3.4 and 5.3.5. First, in Sec. 5.3.4, I derive a set of rewrite procedures that combine
the rewrite rules introduced in 5.1.2 taking into account some relevant properties of SSF.
Then, in Sec. 5.3.5, I introduce a algorithm that applies a compactification procedure to
SSF extended circuits. This algorithm makes use of the rewrite procedures in Sec. 5.3.4
86
to completely rewrite SSF extended circuits until all non-output qubits are removed.
5.3.1
Signal-shifting
Signal-shifting is an optimization technique for measurement patterns able to reduce4
the depth of the computation [35, 23]. The technique is basically an extension of the
measurement calculus rules (Sec. 3.1) which removes from a pattern all Z-corrections
acting on non-output qubits, transferring the dependency to existing X operators. In
what follows, I review the signal-shifting rules and show how it works with an example.
An algorithm that applies the signal-shifting rules to regflow measurement patterns is
given in Sec. 5.3.1.2.
5.3.1.1
Rewrite rules
The signal-shifting rules act by changing the correction dependency of a (standard)
measurement pattern. We start by substituting a S command for a Z command in the
pattern using the following rule:
↵ s
t [Mi ]
) Sit [Mi↵ ]s
(5.8)
where the compact form of the measurement command (Eq. 3.1) was used. Then we
move the S command to the left of the pattern, changing the pattern according to the
following rules:
↵ s
t [Mj ]
Sir ) Sir
↵ s[(r+si )/si ]
t[(r+si )/si ] [Mj ]
(5.9)
s[(t+si )/si ]
(5.10)
s[(t+si )/si ]
(5.11)
Xjs Sit ) Sit Xj
Zjs Sit ) Sit Zj
where Sit is the signal shifting command and s[t/si ] denotes the substitution of si with t
in s. Note that Eq. 5.9 is basically a composition of rules in Eq. 5.10 and 5.11 using the
compact notation for the measurement command (Eq. 3.1).
When the S command reaches the left-most position in the pattern, it is removed.
Here I assume without loss of generality that there is just one S command in the pattern
at a time: a new S command is created only after the existing one has been eliminated.
Moreover, since we are assuming in this thesis that all patterns are in the standard form
4
More precisely, it is a technique that does not increase the depth of the pattern, being able to reduce
it in some cases.
87
(Sec. 3.1.4), we do not need any commutation rule for E and S commands.
As can be seen from the rules above, signal shifting rewrites the X- and Z-corrections
of a measurement pattern in a well defined manner. In particular, it will move all the
Z-corrections to the end of the pattern, thereby introducing new X-corrections when Rule
5.10 is applied. It is proven in [23] that signal shifting will never increase the depth of
the MBQC pattern, although it may decrease it in some cases (see below). In the case
when the depth decreases, it is the consequence of the removal of the Z-corrections on
the measured qubits by applying Rule 5.8.
It is important to note that the S command itself has no physical meaning, being just
a formal procedure that interchanges dependencies in a measurement pattern. Therefore,
a pattern containing S commands will not be physically implemented, since it may even
be non-deterministic due to the incomplete correction structure of the pattern.
Example
Here I give an example of the signal shifting rules being applied to the regflow pattern.
In what follows we start with the pattern in Eq. (3.35) and proceed by applying the rules
being referred to after each arrow ()):
Eq.(5.8)
(5.12)
Free commutation
(5.13)
Z5s3 S3s1 X5s4 s2 [M40 ][M3 ]s2 [M2 ]s1 M1 ↵ E12345 )
Eq.(5.11)
(5.14)
Z5s3 +s1 X5s4
Eq.(5.8)
(5.15)
Eq.(5.10)
(5.16)
Z5s3 X5s4 s2 [M40 ] s1 [M3 ]s2 [M2 ]s1 M1 ↵ E12345 )
Z5s3 X5s4 s2 [M40 ]S3s1 [M3 ]s2 [M2 ]s1 M1 ↵ E12345 )
0
s2 [M4 ]
[M3 ]s2 [M2 ]s1 M1 ↵ E12345 )
Z5s3 +s1 X5s4 S4s2 M40 [M3 ]s2 [M2 ]s1 M1 ↵ E12345 )
Z5s3 +s1 X5s4 +s2 M40 [M3 ]s2 [M2 ]s1 M1 ↵ E12345
(5.17)
Note that in Eq. 5.17 the measurement on qubit 4 does not depend on any measurement outcome. Therefore, the measurement on qubit 4 can be performed simultaneously
with any other measurement in the pattern, hence reducing the depth of the computation.
5.3.1.2
Algorithm
I now present an efficient algorithm (Algorithm 2) for signal shifting a general regflow
pattern (Eq. 5.18). We keep in mind that the order in which we apply the signal shifting
88
rules does not matter [33].
Algorithm 2: SignalShif t
Input: A measurement pattern P with regflow (f, f ) as defined in Equation 5.18.
Output: The signal shifted pattern P 0 of P .
1 begin
2
B = OC ;
3
P0 = P;
4
while B 6= ; do
5
select any vertex i 2 B which is smallest according to f ;
6
B = B \ {i};
7
while 9k 2 B s.t. Zksi 2 P 0 do
8
Move the Zksi command next to the Mk↵k command;
9
Use Rule 5.8 on P 0 to create the signal command Sksi ;
10
// Removes the Zksi command from P 0 ;
11
Use Rule 5.10 on P 0 to create a new Xfsi(k) command;
12
for j 2 N (f (k)) \ {k} do
13
Use Rule 5.11 on P 0 to create a new Zjsi command.
Move Sksi to the end of the pattern and remove it.
14
Proposition 2 (proposition 1 in [42]) Given the measurement pattern P of a regflow
(f,
f)
as defined in Equation 5.18 as input to Algorithm 2, the output will be the signal
shifted measurement pattern of P .
Proof: We will prove this proposition by showing that:
• Algorithm 2 terminates.
• Every step in Algorithm 2 that modifies the pattern P 0 is a valid application of a
signal shifting rewrite rule.
• The output of Algorithm 2, the pattern P 0 , is signal shifted.
We begin by showing that Algorithm 2 will terminate. The first “while” loop will
obviously terminate, as we decrease the number of elements on each loop iteration and
never add anything to the set B. The second “while” loop will not terminate only if some
Zksi command is added to the pattern an infinite number of times. As the underlying
graph is finite and a Zksi command represents a directed edge in the Z-correction graph,
this implies the existence of a cycle in the graph, however this is impossible according to
89
the flow definition. The “for” loop in the algorithm terminates because the graph itself is
finite, hence Algorithm 2 has to terminate.
For Algorithm 2 to actually perform the signal shifting, its operations have to be
either trivial commuting rules or the three signal shifting Rules 5.8, 5.10 and 5.11. As
can be easily seen from the algorithm, the operations done are indeed the signal shifting
rewrite rules. We still need to prove that these rules can be applied in the order shown
in the algorithm. Obviously we can use Rule 5.8 on line 8 to create the signal command
due to the fact that k 2 B ✓ OC and that every non-output qubit is measured. Hence
we have the measurement required for the creation of the signal command in the pattern.
We know that Zksi has to be in the pattern after the command Mi↵i and before Mk↵k . The
entanglement and creation commands are the first commands in the pattern and we do
not need to move the Zksi command past them. Hence we only need to move Zksi past
measurement commands on qubits that are not i and k and other correction commands.
These can be done trivially and hence we can always move the Zksi command next to Mk↵k
to apply Rule 5.8.
Next we want to move the newly created Ski command to the end of the measurement
pattern. To do that we need to commute it past the commands that appear after it.
The only commands Ski commutes non-trivially with are the ones that depend on the
measurement of qubit k as can be seen from Rules 5.8 - 5.11. Those are the X- and
Z-corrections depending on the measurement outcome of qubit k. According to Equation
5.18 there is exactly one such X-correction in the pattern P , namely Xfk(k) . Also the
previous steps of the algorithm could not have created any dependencies from qubit k.
The Z-correction commands have only been created depending on vertices that we already
moved from B. Therefore we need to create exactly one new X-correction command using
Rule 5.10. We also look at the Z-corrections depending on k and from Equation 5.18 we
see that in the original pattern these are on vertices from the set N (f (k)) \ {k}. As for
the X-corrections we also have not created any new Z-corrections from k in the previous
steps of the algorithm. Hence this is exactly the set of corrections we need to commute
with and apply Rule 5.11. We are only left with commands after Ski in the pattern that
commute trivially with Ski . We can move the command to the end of the pattern. The
signal command at the end of the pattern does not influence the computation and we
will not add any new commands to the end of the pattern. Hence we can remove the Sik
command from the pattern.
90
Finally we show that no more signal shifting rules can be applied after the completion
of Algorithm 2, i.e. the pattern P 0 is signal shifted. We eliminate all Z-corrections acting
on a non-output qubit depending on a vertex i after removing it from the set B and will
afterwards never create any new Z-corrections depending on that vertex. At the end of
the algorithm the set B is empty, hence there cannot exist any non-output qubit that
has a Z-correction acting on it and Rule 5.8 cannot be applied anymore. Moreover, since
every signal command is at the end of the pattern, we cannot apply Rules 5.10 and 5.11
either. This completes the proof.
5.3.2
⇤
The signal-shifted flow
Here I introduce and explore several properties of a new type of flow, called signalshifted flow (SSF). The SSF measurement pattern is obtained by applying the signal
shifting rules to a regflow measurement pattern. The signal shifting rules change a pattern’s correction structure, which will allow us to define a flow to the associated graph.
The SSF has several useful properties that we shall see in this chapter - as the existence
of a SSF compactification procedure (given in Sec. 5.3.5) - and in Chapter 6 - where I
use the SSF compactification procedure to construct a circuit optimization scheme.
We shall start by analyzing some properties of the signal shifting rules. The Rules
5.8 - 5.11 can be interpreted in the following way. Signal shifting takes a signal from a
Z-correction on a measured qubit i (Rule 5.8) and adds it to the corrections that depend
on the outcome of the measurement of i (Rules 5.9 - 5.11). If the signal was added to
another Z-correction of a measured vertex, then signal shifting can be applied again until
no Z-corrections are left on non-output vertices. Therefore signals move along a path
created by the Z-corrections. The propagation of signals in an MBQC pattern can be
described by a Z-path as defined below.
Definition 20 (Z-Path) Let M be a measurement pattern on an open graph (G, I, O).
Then we define a directed acyclic graph, called GZ , on the vertices of G such that there
exists a directed edge from i to j iff there exists a correction command Zjsi in M . A path
in GZ between two vertices v and u is called a Z-path.
The above definition allows us to state a simple observation about the connectivity
of a graph with flow.
91
Lemma 3 If (f,
f)
is a flow on an open graph (G, I, O), and there exists a Z-path from
vertex i to vertex j, then the vertices i and f (j) cannot be connected.
Proof. The existence of a Z-path from i to j implies that i
f
j. The Z dependency
graph is an acyclic graph, thus i 6= j. If i were connected to f (j), then according to the
regflow property (F2) j
i
f
j and j
f
f
i. Now we have two contradicting strict partial order relations
i. Therefore i cannot be connected to f (j). ⇤
Recall that the addition of signals is done modulo 2, therefore, if an even number of
signals from a measured vertex i is added to a correction command on vertex j, the signals
will cancel out (since Z 2 = X 2 = I). Furthermore, it is evident from the signal-shifting
rewrite rules that after signal shifting, the measurement result of vertex i will create a
new X-correction over vertex j if there exists an odd number of Z-paths from i to a vertex
k such that j is X-dependent k in the original pattern. Similarly a new Z-correction from
i to j will be created if there exists an odd number of Z-paths from i to j. Either way,
the number of Z-paths from a vertex i to another vertex j, denoted as ⇣i (j), can be used
to determine if the signal from i should be added to a correction. Also, we define ⇣i (i) to
be 1 to simplify further calculations.
Before defining the signal shifted flow (SSF), some definitions and lemmas are needed
to justify our definition. Note that if an open graph (G, I, O) has a regflow (f,
f ),
then
we can write the MBQC pattern of a deterministic computation on this open graph as
[33]:
P =
f ⇣
Y
Xfsi(i) ZNsi(f (i))\{i} Mi↵i
i2OC
where the product follows the strict partial order
f
⌘
(5.18)
E G NI C
of the regflow (f,
f ).
From Eq.
5.18 we see that a Z-correction on a vertex j depending on the measurement outcome of
another vertex i appears only if j is a neighbour of f (i). This is formally stated in the
next corollary as we will refer to it several times.
Corollary 1 If (G, I, O) is an open graph with a regflow (f,
f ),
then there exists a
Z-correction from vertex i to another vertex j iff j 2 N (f (i)) \ {i}.
We define the Z-dependency neighbourhood of a vertex j to be the set of vertices from
which j is receiving a Z-correction. This set has an explicit form given as NZ (j) = {k 2
OC |f (k) 2 N (j) \ {f (j)}}. This is due to the following facts: first, f (k) has to exist for
92
all vertices k 2 OC because of the regflow definition; second, since f (k) 6= f (j) hence k
cannot be equal to j. Because f (k) 2 N (j) ) j 2 N (f (k)) and Corollary 1 there exists
a Z-correction from k to j. It is easy to see, that ⇣i (j) can be written as:
⇣i (j) =
X
(5.19)
⇣i (k).
k2NZ (j)
There exists a Z-correction from every k 2 NZ (j) to j. These Z-corrections can be used
to extend every such Z-path to k to reach j. If i is in the sum, then because ⇣i (i) = 1 the
correct number of Z-paths is obtained with Eq. 5.19.
As mentioned before, the evenness of the number of Z-paths can be used to determine
if a signal is added to a correction command. Let parity(n) be the function that determines
the oddness or evenness of the integer n, i.e. parity(n) = n mod 2. Then if an open
graph has a regflow, the oddness of ⇣i (j) can be found as described in the following lemma.
Lemma 4 (lemma 5 in [42]) For every two vertices i and j in an open graph (G, I, O)
with regflow (f,
f)
parity(⇣i (j)) = |k 2 {NZ (j)|parity(⇣i (k)) = 1}|
mod 2
i.e. parity(⇣i (j)) depends only on the number of vertices in the Z-dependency neighbourhood which have odd number of Z-paths from i.
Proof: The oddness of ⇣i (j) can be written as
0
1
X
parity(⇣i (j)) = @
⇣i (k)A mod 2 =
k2NZ (j)
=
X
k2NZ (j)
=
(⇣i (k)
X
{k2NZ (j)|1=⇣i (k)
mod 2)
mod 2 =
(⇣i (k)
mod 2)
mod 2 =
mod 2}
= |{k 2 NZ (j)|1 = ⇣i (k)
mod 2}|
= |{k 2 NZ (j)|parity(⇣i (k)) = 1}|
mod 2 =
mod 2
⇤
All these notions will allow us to define the structure of the pattern after signal shifting
has been performed.
Proposition 3 (proposition 2 in [42]) Given a regflow (f,
f)
on an open graph
93
C
(G, I, O), let s be a function from OC 7! P I such that j 2 s(i) iff parity(⇣i (f
1
(j))) = 1.
Also define Ls to be a layering function from V (G) into a natural number:
Ls (i) = 0
8i 2 O
Ls (i) = max (Ls (j) + 1)
8i 2
/O
j2s(i)
Define the strict partial order
s
i
with:
s
j
,
Ls (i) > Ls (j)
Then, the application of signal shifting Rules 5.8 - 5.11 over an MBQC pattern with
regflow (f,
f)
will lead to the following pattern:
P =
Y
j2O,i2I C
s .parity(⇣i (j))
Zj i
s ⇣
Y
i2OC
⌘
si
Xs(i)
Mi↵i EG NI C
(5.20)
Proof: The proof is divided into three parts. First we will show that signal shifting creates
exactly the pattern commands shown in Equation 5.20. We proceed by showing that the
layering function Ls is defined for every i 2 V (G). Lastly, we need to prove that using
the partial order
s
derived from Ls for ordering the commands as in Equation 5.20 gives
a valid measurement pattern.
Note that the preparation commands (NIC ), entanglement commands (EG ) and measurement commands (Mi↵i ) are the same for Equations 5.18 and 5.20. Because signal
shifting does not change these commands (Rules 5.8 - 5.11) these are as required for a
signal shifted pattern. Hence we need only to consider the correction commands.
We will look at the correction commands that would appear in a signal shifted pattern.
We do this by examining the signal shifting algorithm (Algorithm 2). As mentioned before,
the algorithm works as a directed graph traversal, in a way that every distinct path is
traversed. As seen in the algorithm every Zksi correction acting on a non-output qubit is
removed from the pattern. This is in accordance with the proposed pattern in Equation
5.20. Let us examine which new corrections are created.
The number of newly created Xjsi depends on the number of times we enter the first
loop with command Zfsi 1 (j) . As the algorithm is a directed graph traversal algorithm, this
happens as many times as there are different paths over the Z-dependency graph from i to
f
1
(j). Because the same two Xlsi corrections cancel each other, hence a new X-correction
appears in a signal shifted pattern only if parity(⇣i (f
new
Xfsi(i)
1
(j))) = 1. We also note that no
correction is created since there exist no Z-path between i and f
1
(f (i)). On
94
the other hand Algorithm 2 leaves the already existing X corrections unchanged and
moreover since we have defined ⇣i (i) = 1 therefore f (i) 2 s(i). This implies that the set
s(i) does indeed contain all the vertices that have an X-correction depending on si after
signal shifting is performed.
The number of newly created Z-corrections on an output vertex j depending on a
vertex i appearing in the signal shifted pattern is equal to the number of different paths
from i to j. The difference with non-output qubits is that these will not be removed
through the process of signal shifting. As with X-corrections, two Z-correction commands
on the same qubit will cancel each other out and hence the existence of a Zjsi in the final
pattern depends on the parity of the number of paths from i to j. This can be written in
short as:
s ·parity(⇣i (j))
Zj i
Hence the measurement pattern in Equation 5.20 has exactly the same commands as the
signal shifted pattern in Equation 5.18.
Another thing we need to prove is that the layering function Ls is defined for every
i 2 V (G). As proven above, the X-corrections depending on the measurement of qubit v
correspond to the set s(v). Hence we can interpret the definition of Ls (v) as finding the
maximum value of Ls for every vertex that has an X-correction from v and adding 1 to
it. The recursive definition of Ls (v) is well defined, if for every non-output qubit we can
find a path over X-corrections ending at an output qubit. We know that signal shifting of
a valid pattern creates another valid pattern. This implies that the X-corrections cannot
create a cyclic dependency structure and hence every path over the X-corrections has an
endpoint. Moreover such a path cannot end on a non-output qubit k since f (k) 2 s(k)
and one could always extend that path with f (k). Therefore Ls (v) is well defined.
Finally, it is easy to show that the partial order
s
as used in Equation 5.20 gives
a valid ordering of the commands. Every vertex j that has an X-correction depending
on the measurement of qubit i has a smaller Ls number and hence i
s
j. This way no
X-correction command acts on an already measured qubit and because the Z-corrections
are applied only on output qubits, the correction ordering is valid. Every other command
is applied before the measurement command and hence the pattern in Equation 5.20 is a
valid measurement pattern.
⇤
Given an open graph with regflow, we refer to the construction of the above proposition
as its corresponding signal-shifted flow (SSF).
95
Corollary 2 (corollary 2 in [42]) If (G, I, O) is an open graph with regflow (f,
and SSF (s,
s)
f)
then for every vertex i and j such that f (j) 2 s(i) \ {f (i)}, we can find
another vertex k, such that f (k) 2 s(i) \ N (j).
Proof: If f (j) 2 s(i), then from the Proposition 3 of SSF we can conclude that
parity(⇣i (j)) = 1. We know that j 6= i from the assumptions. Lemma 4 says that
there must exist at least one other vertex k from which j has a Z-correction, such that
parity(⇣i (k)) = 1. The regflow definition says that j must therefore be a neighbour of f (k).
Definition 3 of SSF states that f (k) must therefore be in s(i), hence f (k) 2 s(i) \ N (j).
⇤
The SSF has several interesting properties, but two of them are specially relevant for
this thesis. Both properties were proved in [42] and are stated in the theorems below. The
proofs were omitted since they are rather long and are out of the scope of this thesis. The
two SSF properties stated in the theorems below have motivated the developement of a
SSF compactification procedure (Sec. 5.3.5) and the circuit optimization scheme analyzed
in Chapter 6.
Theorem 2 (theorem 1 from [42]) Given any open graph (G, I, O) with flow (f,
the corresponding signal shifted regflow (s,
s)
is a gflow.
Theorem 3 (theorem 2 from [42]) Let (G, I, O) be an open graph with regflow (f,
such that |I| = |O|. Let (s,
s)
f ),
be the SSF obtained from (f,
f ).
Then (s,
s)
f)
is the
optimal gflow of (G, I, O).
Therefore, if a graph has regflow, the SSF provides the optimal correcting strategy
(with respect to the depth of the pattern). Moreover, the SSF flow is easier to find than
the optimal gflow introduced in [81], specially due to the fact that SSF pattern can be
constructed by the application of the simple rewrite rules in Eqs. 5.8 to 5.115 .
5.3.3
SSF structural properties
The notions of influencing walks and partial influencing walks on open graphs with
flow were introduced in [23] to describe the set of all vertices that a measurement depends
5
Note, however, that it is valid only for graphs with regflow. Since SSF is not defined for graphs that
do not satisfy regflow conditions, the method of [81] is the only known method for finding the optimal
gflow in those cases.
96
on. An influencing walk starts with an input and ends with an output vertex, a partial
influencing walk starts with an input vertex but can end with a non-output vertex. We
will use a modified definition of influencing walks that can start from any non-output
vertex i and end at any vertex j 2 s(i) and call it a stepwise influencing path. This will
allow us to conveniently explore the dependency structure of a pattern with SSF.
Definition 21 Let (s,
s)
be an SSF that is obtained from a regflow (f,
f)
of an open
graph (G, I, O) and vertices i and j in V (G) such that j 2 s(i). We say that a path
between vertices i and j is an stepwise influencing path, noted as }i (j), iff
• The path is over the edges of G.
• The first two elements on the path are i and f (i).
• Every even-placed vertex k on the path }i (j), starting from f (i), is in s(i).
• Every odd-placed vertex on the path }i (j) is the unique vertex f
such that k is the next vertex on the path }i (j).
1
(k) of some k 2 s(i)
s(i)
i
-1
f (k2)
.
.
.
-1
f (kn)
f(i)=k1
k2
.
.
.
j=kn
Figure 22: Stepwise influencing path }i (j). See Def. 21.
It is easy to see that every second edge, in particular the edges between f
1
(k) and
k 2 s(i), in the stepwise influencing path is a flow edge. Hence the path contains no
97
consecutive non-flow edges. If we restrict the first vertices of the stepwise influencing path
to be input vertices, the stepwise influencing path would be a partial influencing path,
but not vice versa. Stepwise influencing paths are useful because of their appearance in
the SSF as proven by the following lemma.
Lemma 5 Let (s,
s)
be an SSF obtained from a regflow (f,
f)
of an open graph (G, I, O)
and vertices i and j in V (G) such that j 2 s(i). Then there always exists a stepwise influencing path }i (j).
Proof: We start by constructing such a path backward from j to i. We select j and
f
1
(j) as the last two vertices on the path and apply Corollary 2 to find the vertices on
the path, until we reach i. The formation of cycles is impossible, as this would imply a
cyclic dependency structure, impossible for a regflow. We have to reach i as the set of
⇤
vertices we choose from is finite.
Note that there might be more than one stepwise influencing path from i to j. Next,
I show that the structure of stepwise influencing paths imposes a strict restriction on the
way a vertex on the stepwise influencing path can be connected.
Lemma 6 Let }i (j) be a stepwise influencing path from i to j in an open graph (G, I, O)
with flow (f,
f)
and corresponding SSF (s,
s ).
Then f
1
(j) is the only odd-placed vertex
in }i (j) that j is connected to.
Proof: According to the definition of stepwise influencing path, for every three consecutive
vertices v1 , v2 , v3 in }i (j) such that v1 and v3 are odd-placed we have that v2 = f (v1 ) and
v3 2 N (v2 ) = N (f (v1 )). According to Corollary 1 there must exist a Z-correction from
v1 to v3 . Therefore the odd-placed vertices in }i (j) are on a Z-path from i to f
obviously from every odd-placed vertex in }i (j) there exists a Z-path to f
1
(j) and
1
(j). Lemma
3 says that j cannot be connected to any of the odd-placed vertices in }i (j).
⇤
The previous lemma shows that the stepwise influencing paths can be used to describe
some properties of the connectivity in open graphs with SSF. The next lemma (illustrated
in Figure 23) will explain how a stepwise influencing path can be extended.
Lemma 7 Let (G, I, O) be an open graph with regflow (f,
(s,
s)
f)
and corresponding SSF
and let i and j be two non-output vertices of the open graph such that f (j) 2 s(i).
If v 2 N (j) \ s(i) \ {f (j)} then every stepwise influencing path }i (v) can be extended by
the vertices j and f (j) to create another stepwise influencing path }i (f (j)).
98
Proof: Adding j and f (j) to }i (v) satisfies the conditions for stepwise influencing paths.
There exists an edge between vertices j and v and vertices j and f (j), hence it is a valid
path. Moreover, f (j) 2 s(i) would be an even-placed vertex on the extended path, and j
would be the unique oddly-placed vertex with f (j) 2 s(i).
⇤
s(i)
i
j
f(i)
initial stepwise
influencing
path
v
extending the
stepwise
influencing
path
f(j)
Figure 23: Extending a stepwise influencing path ending at vertex v according to Lemma
7.
The above lemmas will be used in Section 5.3.5 to obtain compact circuits from SSF.
5.3.4
Rewrite procedures for signal-shifted flow
In what follows we present the rewrite procedures (RPs) that will be used later in the
compactification procedure (Algorithm 4). We refer to the i wire of each RP as the target
wire, j1 , ..., jn as the correcting wires and finally k as the neighbour wire. Moreover, when
we need to emphasise which RP we are referring to we also add a superscript to the wire
label; for instance k (2) indicates the neighbour wire of RP2.
1. Rewrite Procedure 2. The rewrite procedure in Figure 24 moves gates (Ekj1 , ..., Ekjm )
past gates (CXij1 , ..., CXijm ), adding m many gates CZik to slice C in the process.
Using the rewrite rule in Figure 20-d, each of those E gates can be moved past gates
(CXij1 , ..., CXijm ) in C, creating a new CZik each time the rule is applied (Figure
24-c).
99
i
J
i
z
...
J
i
z
...
j
j
j
.
.
.
.
.
.
.
.
.
1
1
jm
jm
...
k
k
z
1
jm
...
J
...
k
(b)
(a)
(c)
Figure 24: Rewrite Procedure 2. This RP is composed by several applications of the
rewrite rule in Figure 20-d. Although all E gates are drawn in the E slice, the requirement
for this RP to be applied is that those E gates are placed just before the corresponding
CX gates (that is, with no gate in between).
2. Rewrite Procedure 3. The rewrite procedure in Figure 25 replaces gates (Ekj1 , ..., Ekjn 1 )
in slice E0 with (CXj1 jn , CXj2 jn , ..., CXjn
1 jn
) in slice C, removing gate CXijn in the
process. We use the rewrite rule in Figure 20-b for each pair {(Ej1 k Ekjn ), ..., (Ejn
transforming gates (Ekj1 , ..., Ekjn 1 ) into (CXj1 jn , ..., CXjn
1 jn
1k
Ekjn )}
), as depicted in Fig-
ure 25-b. The new CX gates can be pushed forward to the beginning of slice C,
since it commutes trivially with CXkjn . Using the rewrite rule in Figure 20-e we can
commute (CXj1 jn , ..., CXjn
1 jn
) past (CXij1 , ..., CXijn 1 ) creating (n
1) many new
CXijn in the process, which together with the pre-existing CXijn in C will cancel
out (since n is even), resulting in the circuit depicted in Figure 25-c.
...
J
i
j , ..., j
n-1
...
...
J
z
n-1
J
z
z
n-1
(n-1)
1
k
J
J
J
f(k)= j n
(a)
(b)
(c)
Figure 25: Rewrite Procedure 3. If Ls (k) = Ls (i), slices (E, J, C) and (E0 , J0 , C0 ) become
the same; The rewrite procedure remains exactly the same.
3. Rewrite Procedure 4. The rewrite procedure in Figure 26 replaces gates (Ekj1 , ..., Ekjn )
in slice E0 with (CXj1 f (k) , ..., CXjn f (k) ) in slice C. We use the rewrite rule in Figure
20-b for each pair {(Ej1 k Ekf (k) ), ..., (Ejn k Ekf (k) )} transforming gates (Ej1 k , ..., Ejn k )
into (CXj1 f (k) , ..., CXjn f (k) ) (Figure 26-b). The new CX gates can be pushed forward to the beginning of slice C, since it commutes trivially with CXkf (k) . Using the rewrite rule in Figure 20-e we can commute (CXj1 f (k) , ..., CXjn f (k) ) past
(CXij1 , ..., CXijn ), creating n many CXif (k) in the process. Since n is even, all
100
those CX gates will cancel, resulting in the circuit depicted in Figure 26-c.
...
...
...
J
i
j , ..., j n
( -1)
1
2
jn, ..., jn
J
z
J
z
n\2
n\2
n\2
n\2
n\2
n\2
z
2
k
J
J
J
f(k)
(a)
(b)
(c)
Figure 26: Rewrite Procedure 4. If Ls (k) = Ls (i), slices (E, J, C) and (E0 , J0 , C0 ) become
the same; The rewrite procedure remains exactly the same.
In order to apply the J-gate identity for the pair of wires (i, f (i)) of a SSF extended
circuit, we need to rewrite it until all conditions in Definition 17 are satisfied. In a SSF
extended circuit, the first two conditions are trivially satisfied for any pair of wires (i, j) =
(i, f (i)) and, therefore, we need to rewrite the circuit to satisfy the other conditions. To
do so we analyse each qubit k in the neighbourhood of s(i), classifying it according to
three different cases: (i) Ls (k) < Ls (i), (ii) Ls (k)
Ls (k)
Ls (i) and f (k) 2 s(i) and (iii)
Ls (i) and f (k) 2
/ s(i). This separation into cases is necessary for two reasons:
First, the distinction between Ls (k) < Ls (i) and Ls (k)
Ls (i) is necessary because
we are interested in keeping the J-gate parallelization introduced by signal shifting and
hence we need a different procedure to deal with each case. Secondly, in the case where
Ls (k)
Ls (i), we use the rewrite rule in Figure 20-b which deletes E gates. Since
condition (ii) in Definition 17 requires the existence of gates of form Ekf (k) , we will treat
differently cases where f (k) 2 s(i) and f (k) 2
/ s(i) to guarantee those E gates will not be
removed from the circuit. As we show next, for each case one of the RPs can be applied
if a set of prior conditions are satisfied.
Proposition 4 Let i be a wire in a SSF extended circuit s.t. Ji is in some slice Jn . If
there exists a wire k s.t. (i) Ls (k) < Ls (i) and (ii) the set of gates {Ekj1 , ..., Ekjm } (with
j1 , ..., jm 2 s(i)) can be pushed to slice Jn , then RP2 (Figure 24) can be applied.
Proof. Since Ls (k) < Ls (i), the gate Jk belongs to a future slice Jm (m > n). Also,
j1 , ..., jm 2 s(i) implies i
s
{j1 , ..., jm } and hence the gates Jj1 , ..., Jjm are in slices after
Jn as well. Moreover, j1 , ..., jm 2 s(i) implies there exist operators Xjs1i , ..., Xjsmi in the
measurement pattern, which are translated to the extended circuit as a CXij1 , ..., CXijm
in slice cn,i (according to Definition 14). Thus, if every gate Ekj , j 2 s(i), can be trivially
101
pushed to slice Jn , we have exactly the scenario depicted in Figure 24-a. Therefore, the
rewrite procedure in Figure 24 can be applied. ⇤
For reasons that will become clear in Sec. 5.3.5, where the algorithm to obtain compact
circuits from SSF extended circuits is introduced, we need to consider a case which is
slightly different from the scenario described in Proposition 4. In this case, condition (ii)
in Proposition 4 is not satisfied because some of the E gates are in slice cn,i but cannot
be pushed trivially back to Jn . The only scenario where that could happen is if there
exists CXi,k in slice cn,i and some of the Ekj gates are placed past it (and hence cannot
be pushed trivially to Jn ). In this scenario, as we show next, RP2 can also be applied.
Proposition 5 Let i be a wire in a SSF extended circuit s.t. Ji is in some slice Jn . If
there exists a wire k s.t. (i) k 2 s(i) and (ii) gates {Ekj1 , ..., Ekjm } (with j1 , ..., jm 2 s(i))
can either be all pushed to slice Jn or just some can be pushed to slice Jn and the other E
gates are placed in cn,i just after CXik , then RP2 (Figure 24) can be applied.
Proof. The proof of this proposition is simple due to its similarity to Proposition 4.
First, note that k 2 s(i) implies Ls (k) < Ls (i) and hence condition (i) in this proposition
is equivalent to the one in Proposition 4. Therefore, if all gates in the set {Ekj1 , ..., Ekjm }
can be pushed to slice Jn we have exactly the conditions in Proposition 4 and there is
nothing to prove. The other possibility is when a subset of the set {Ekj1 , ..., Ekjm } can
be pushed to slice Jn but gates in the complementary subset are placed in slice cn,i , after
gate CXik (which exists since k 2 s(i)). It is easy to note that it will not prevent the
application of RP2, since the only gate in cn,i that could be placed in between the E and
CX gates used in RP2, namely CXik , is assumed in condition (ii) to be placed before the
gates of the form Ekj . ⇤
In the next Lemma we show the interesting effect of applying RP2 to SSF extended
circuits.
Lemma 8 The application of RP2 to a pair of target and neighbour wires (i, k) of a SSF
extended circuit removes all CZik from the circuit.
Proof. According to Definition 14, all CZ gates with control in wire i are placed in
slice cn,i and hence we only need to show that all CZik in that slice are removed. Let us
divide the analysis into two cases: (i) k 2 Odd(s(i)) and (ii) k 2
/ Odd(s(i)). Consider
the first case. Since k 2 Odd(s(i)), there exists a Zksi in the measurement pattern and
102
i
J
z
j
i
J
Z
j
(a)
z
i
J
z
j
(b)
(c)
Figure 27: Removing even many CZij from a SSF extended circuit. See Lemma 8 for
more information.
hence there exists a gate CZik in slice cn,i . For this case, the index m in RP2 is an odd
number and therefore the application of RP2 creates odd many CZik gates in slice cn,i .
Note that those CZik gates can be created in different parts of cn,i (depending whether
Proposition 4 or 5 is satisfied) such that they cannot be grouped together trivially (like
in Figure 27-a). It is easy to verify that the only possible scenario for that to happen is
when there exists CXik in cn,i in between the created CZik gates. However, those CZik
gates can be grouped together by moving all CZik to one side of the troublesome CXik
using the identity
CZik CXik | ii | ik = CXik CZik (Zi | ii )| ik
(5.21)
where | i and | i are arbitrary quantum sates and Z is the Pauli operator (see Figure
27-b). This way, since there exist even many CZik in sequence and even many CZ gates
equal the identity, all CZik can be simply removed from the circuit. The created singlequbit gate Zi can be pushed forward to the end of the i wire, since it commutes with all
gates in cn,i . Moreover, since the final measurement is onto the Z-basis, the operator Zi
has no effect in the measurement statistics and therefore can be removed from the circuit
without changing the computation being implemented by the circuit (Figure 27-c). An
equivalent analysis applies for the second case, where k 2
/ Odd(s(i)). For this case RP2
create even many CZik in slice cn,i and there is no pre-existing CZik in that slice. The
same identity can be used to group together all created CZik , which cancel out since there
are even many of them. Since both created and pre-existing CZik are removed from the
circuit by the application of RP2, the Lemma holds. ⇤
Now we analyse the cases where Ls (k)
Ls (i), that is, with Jk gate blocking all gates
of form Ekj to be pushed past the correction gates (necessary to satisfy condition (iii) in
Definition 17). Since in those cases the corresponding RP will delete E gates and we do
not want to delete E gates of the form Ei,f (i) (see Definition 17), we analyse separately
the case where f (k) 2 s(i) (Proposition 6) and the case where f (k) 2
/ s(i) (Proposition
7).
103
Proposition 6 Let i be a wire in a SSF extended circuit s.t. Ji is in slice Jn . If there
exists a wire k such that the following conditions are satisfied: (i) Ls (k)
Ls (i); (ii)
f (k) 2 s(i); (iii) gates Ekj1 , ..., Ekjn (with j1 , ..., jn 2 s(i)) can be trivially pushed to some
slice Jn0 (containing Jk ); and (iv) the Ekf (k) gate is the first gate acting on wire f (k),
then RP3 (Figure 25) can be applied.
Proof. Since Ls (k)
Ls (i), n0  n. In slice Cn0 there is a CXkf (k) ; since f (k) 2 s(i),
we define jn = f (k). Also, j1 , ..., jn 2 s(i) implies that i
s
{j1 , ..., jn } and hence the
gates Jj1 , ..., Jjn are in some future slices (compared to n). Moreover, it also implies the
existence of the operators Xjs1i , ..., Xjsni in the measurement pattern, which are translated
to the extended circuit as CXij1 , ..., CXijn in slice cn,i (containing all the CX and CZ
with control on qubit i). On the other hand, Ls (k)
Ls (i) implies both k 2
/ Odd(s(i))
and k 2
/ s(i) and therefore there is no CZik or CXik in the circuit. Thus, if every gate
Ekj1 , ..., Ekjn can be trivially pushed to slice Jn0 and Ekf (k) is the first gate acting on wire
jn , we have exactly the scenario depicted in Figure 25-a. Therefore, the circuit identity
in Figure 25 can be applied. ⇤
Proposition 7 Let i be a wire in a SSF extended circuit s.t. Ji is in slice Jn . If there
exists a wire k such that the following conditions are satisfied: (i) Ls (k)
Ls (i); (ii)
f (k) 2
/ s(i); (iii) gates Ekj1 , ..., Ekjn (with j1 , ..., jn 2 s(i)) can be trivially pushed to slice
Jn0 (containing Jk ); and (iv) the Ekf (k) gate is the first gate acting on wire f (k), then
RP4 (Figure 26) can be applied.
Proof. Since Ls (k)
Ls (i), then n0  n. In slice Cn0 there is a CXkf (k) , with f (k) 2
/ s(i).
Also, j1 , ..., jn 2 s(i) implies that i
s
{j1 , ..., jn } and hence the gates Jj1 , ..., Jjn are also
in some future slices (compared to n). Moreover, it also implies the existence of the
operators Xjs1i , ..., Xjsni in the measurement pattern, which are translated to the extended
circuit as CXij1 , ..., CXijn in slice cn,i (containing all the CX and CZ with control on
qubit i). On the other hand, Ls (k)
Ls (i) implies both k 2
/ Odd(s(i)) and k 2
/ s(i) and
therefore there is no CZik or CXik in the circuit. Thus, if every gate Ekj1 , ..., Ekjn and
Ekf (k) can be trivially pushed to slice Jn0 and Ekf (k) is the first gate acting on wire f (k),
we have exactly the scenario depicted in Figure 26-a. Therefore, the circuit identity in
Figure 26 can be applied. ⇤
In what follows we analyse some properties regarding the interplay between the RPs
that will be crucial for the compactification algorithm for SSF extended circuits.
104
Lemma 9 Let C be an extended circuit obtained from a signal shifted measurement pattern. Then, if a rewrite procedure removes a gate Ejk from the circuit, it will not be
required for the application of any other rewrite procedure.
Proof. Note that only RP3 and RP4 can delete a Ejk gate from the circuit. The
proof is divided into three parts: (i) we show that the Ejk is never a flow edge and,
therefore, it will not be required for the application of a J-gate identity; (ii) it will not be
needed for the application of RP2 and (iii) it will not be needed to create a CX (by using
RP3 or RP4) in the next layers. Suppose Ejk is a flow edge such that k = f (j), with
j being any of qubits (j1 , ..., jn 1 ) in RP3 or (j1 , ..., jn ) in RP4. The other way around,
that is, j = f (k), is not possible since f (k) is separately identified in both RP3 and RP4.
Since f (j) 2 s(j) for all j 2 OC , we have j
s
k. By construction, RP3 and RP4 are
applied if and only if k 4s i. Moreover, since j 2 s(i) in both RP3 and RP4, it holds that
i
s
j. Putting everything together we have i
s
j
s
k 4s i, which is a contradiction.
Therefore, Ejk is never a flow edge.
In this second part of the proof, we show that if Ejk is deleted by RP3 or RP4, it will
not be required for the application of RP2 in any future step. Using the notation defined
earlier, the fact that RP2 is applied after RP3 or RP4 can be written as i(3)
i(4)
s
s
i(2) or
i(2) , where i is the target wire of the corresponding RP. This relation will be used
(2)
(2)
in the rest of the proof. We show that k (3) and k (4) cannot be any of wires j1 , ..., jn
(denoted simply by j (2) in the rest of this proof) or k (2) and hence the gates Ej (3) k(3) or
Ej (4) k(4) cannot be the same as Ej (2) k(2) , which is what we want to prove. Let us show for
k (3) . Since k (3) 4s i(3) and i(2)
s
k (2) (true for both Propositions 4 and 5, which state
the scenarios where RP2 is applied) we have k (3) 4s i(3)
s
i(2)
s
k (2) and therefore k (3)
cannot be the same wire as k (2) . Now assume wire j (3) is k (3) . In RP4 we have k (4) 4s i(4)
(by construction) and in RP2 it holds that i(2)
s
j (2) , since the target of a correcting
gate is always placed before the J-gate acting on that same wire (Definition 14). Putting
everything together gives k (3) 4s i(3)
it gives k (3)
s
s
i(2)
s
j (2) . But since by assumption j (2) is k (3) ,
k (3) which is a contradiction. The proof is the same for k (4) . Therefore, if
Ejk is “consumed” by RP3 or RP4, it will not be required for the application of RP2 in
any future step.
Finally for the third part note that an E gate will be removed to create a CX gate
only through identity in Figure 20-b. The pair of such E gates is always associated with
a flow and a non-flow edge when used in RP3 or RP4. Therefore, there are two possible
pairs of E gates using the non-flow edge Ejk : Ejk Ekf (k) (which would create CXjf (k) ) and
105
Ejk Ejf (j) (which would create CXkf (j) ). We want to show that whenever we “consume”
Ejk , it will not be required to create a different CX than the one already created. Suppose
we use Exy to create CXxf (y) . This is the case when y 4s i and i
s
x, if x is a correcting
wire and y is a neighbour wire in either RP3 or RP4. Similarly, if x is the neighbour
wire and y is the correcting one then x 4s i and y
s
i (easily obtained by relabelling
the wires in RP3 and RP4). Since the first pair of conditions is inconsistent with the last
pair of conditions (j 4s i and k
s
i), we conclude that once a given E is used to create
a CX, that E would not be required to create a different CX in any further step of the
algorithm since the partial order
s
is not changed by the application of any RP. ⇤
Corollary 3 Let CXij be a gate in an extended circuit created by the application of a
rewrite procedure. The E gate “consumed” to create this CXij can be univocally determined.
Proof. It follows from the third part of the proof of Lemma 9 that if a gate of the form
Ejk is deleted in RP3 or RP4 we will have one of the following cases: the gates Ejk Ekf (k)
are used to create CXjf (k) or the pair Ejk Ejf (j) are used to create CXkf (j) . The CX that
is created by “consuming” the Ejk depends exclusively on the relation between vertices j,
k and f (j) in respect to the partial order
s.
Since the partial order
s
is not changed
by the application of any RP, there is a one-to-one correspondence between the deleted
E gate and the newly created CX gate. ⇤
The process of choosing the correct RP to be applied is summarized in Algorithm 3,
which will be used as a subroutine in the compactification algorithm for SSF extended
circuits (Algorithm 4). The modified conditions in Lines 5 and 7 are based on the fact
that if a CZ required for the application of RP3 or RP4 is not in the circuit, the CX
that it would create will be (see Lemma 9), allowing the application of future RPs. In
all RPs we assumed that some E gates could be moved to the beginning of a given slice
J in a trivial way. It will not be true in general for SSF extended circuits, since there
might exist several other gates in the extended circuit such that the aforementioned E
gates could not be moved trivially to J. In other words, the conditions in Propositions
4 to 7 would not be satisfied. The algorithm in the next section addresses exactly this
problem: it provides an ordering where Algorithm 3 never aborts. This ordering is the
global structure coming into play, since it is related to SSF and flow of the graph.
106
Algorithm 3: This algorithm decides which Rewrite Procedure must be applied by
analysing how a given qubit k is connected with the correcting set of another qubit
i
Input: Wire labels i and k.
Output: A description of which Rewrite Procedure must be applied.
1 begin
2
Read i and k.
3
if all conditions in Proposition 4 or Proposition 5 are satisfied then
4
Apply RP2;
if all conditions in Proposition 6 are satisfied or conditions (iii) and/or (iv) in
Proposition 6 are not satisfied but there exist a sequence of
(CXj1 jn , ..., CXjn 1 jn ) gates that can be pushed trivially to cn,i then
Apply RP3;
5
6
8
if all conditions in Proposition 7 are satisfied or conditions (iii) and/or (iv) in
Proposition 7 are not satisfied but there exist a sequence
(CXj1 f (k) , ..., CXjn f (k) ) gates that can be pushed trivially to cn,i then
Apply RP4;
9
otherwise Abort
7
5.3.5
Compactification procedure for signal-shifted flow
In this section I explain how the compactification algorithm for SSF extended circuits
(Algorithm 4) works. The goal of Algorithm 4 is to create |OC | many J-blocks (see
Definition 17) in a SSF extended circuit and then apply the J-gate identity for all Jblocks, removing |OC | many wires from the extended circuit. Therefore, the output of
Algorithm 4 is the compact form of the inputed SSF extended circuit. One of the main
differences between SSF and other gflows is the notion of stepwise influencing paths,
introduced in Section 5.3.3. The next lemma uses some properties of stepwise influencing
paths to relate the SSF correcting set to the partial order of regflow. This relation will play
an important role in Algorithm 4 since it gives an appropriate ordering for the application
of the RPs.
Lemma 10 Let (G, I, O) be an open graph with regular flow (f,
Then, for all v 2 N (j) \ {i}, where j 2 s(i), it holds that v
f
f)
and SSF (s,
i.
Proof. First suppose j = f (i). Then, by regflow definition (Def. 10), v
suppose j 6= f (i). Then from Def. 10, v
f
influencing path passing through f (i) and f
2, that there exists f
f
1
(j)
f
f
1
1
1
f
1
s ).
f
i. Now
(j). By Lemma 5, there exists a step-wise
(j), namely }i (j). It follows from Corollary
(k) such that k 2 s(i) and f
1
(j) 2 N (k) and, consequently,
(k). Lemma 7 allows us to repeat this process to find the previous vertices
107
in the path }i (j) until we reach vertex i. Hence we conclude that f
v
f
f
1
(j)
f
1
(j)
i
i and therefore
i. ⇤
Before running Algorithm 4, the slices cn,i must be arranged in the extended circuit
from right to left respecting the order imposed by
f.
This can always be done since
the set of gates in a given slice cn,i commutes with the gates in an other slice cn,j , for
any valid j. The algorithm starts with a for loop that runs for one SSF layer at a time,
starting with the input layer and moving onwards until the last layer of non-output qubit
is considered. At each iteration n of the for loop, a rewrite procedure is applied to each
pair of qubits (i, k) s.t. Ji is in Jn and k 2 N (s(i)). The two foreach loops and the two
while loops define the order in which Algorithm 3 will be called. This ordering, which is
the inverse of the order given by
f,
assures that Algorithm 3 will never abort when it is
called by Algorithm 4.
Algorithm 4: Transforms a SSF extended circuit into a compact circuit.
Input: SSF extended circuit.
Output: Compact form of the SSF extended circuit.
1 begin
2
D
maxi2G {Ls (i)}
3
for n = 1 to D do
4
Let Wn = {an,1 , ..., an,mn } be the set of the mn wires with J-gate in slice Jn
n
5
Let Smax
maxi2Wn {Lf (i)}
n
6
Let Smin
mini2Wn {Lf (i)}
n
n
7
while Smax
Smin
do
n
8
foreach an,i 2 Wn such that Lf (an,i ) = Smax
do
9
Let N ST EP
maxk2{N (s(an,i ))\{an,i }} {Lf (k)}
n
10
while N ST EP > Smax
do
11
// That is, for all k 2 {N (s(an,i )) \ {an,i }} (Due to
Lemma 10)
12
foreach k 2 {N (s(an,i )) \ {an,i }} such that Lf (k) = N ST EP
do
13
i
an,i ;
14
Run Algorithm 3
N ST EP
15
16
17
18
Smax
(Smax
(N ST EP
1)
1)
Remove from the circuit all gates CXij placed in any slice Cn such that
j 6= f (i).
Apply the J-gate identity for all pairs of wires {i, f (i)} such that i 2 OC .
Lemma 11 Algorithm 3 never aborts when called by Algorithm 4.
108
Proof. Let us start by showing that the algorithm works for the first iteration of the
for loop (i.e., n = 1) and then we explain why it will work for all other layers. In what
follows we use the notation an,i to represent a target wire i in layer n; note that there
n
n
might exist more than one such wire in the same layer. Let Smax
(Smin
) be the maximum
(minimum) value of Lf (an,i ) for an,i 2 Wn (as defined in Lines 5 and 6 in the algorithm),
where Wn is the set of the mn wires with J-gate in slice Jn . According to Lemma 10,
1
for any a1,i such that Lf (a1,i ) = Smax
, and any qubit k connected to a qubit in s(a1,i ) it
holds that k 2
/ W1 . In this case, condition (i) in Proposition 4 is satisfied. Since we are
in the first SSF layer, all E gates required for the application of any of those RPs can
be trivially pushed forward to slice J1 , hence all conditions in Proposition 4 are initially
satisfied. After the application of one or more RP2, it might happen that some E gates
get stuck in between two CX gates in slice c1,i (which happens when a neighbour wire k
is itself in both N (s(a1,i )) and s(a1,i ) sets). In this case the conditions in Proposition 5
are then satisfied. In both cases, only RP2 can be selected by Algorithm 3. Therefore,
1
Algorithm 3 would not abort for all target qubits a1,i such that Lf (a1,i ) = Smax
. As a
consequence, since only RP2 is applied, all gate Ekp , such that p 2 s(a1,i ), are moved past
1
the correction slice c1,i , for all a1,i such that Lf (a1,i ) = Smax
.
1
Now consider qubits a1,i such that Lf (a1,i ) = Smax
1, that is, the second iteration
of the first while loop. For k 2 N (s(a1,i )), Lemma 10 implies that either k 2
/ W1
1
or {(k 2 W1 ) ^ [Lf (k) = Smax
]} is true. For k 2
/ W1 , the procedure to be applied
is RP2, similarly and by the same reasons as in the previous iteration. For the other
case the applicable rewrite procedures are either RP3 or RP4, since Jk in J1 implies
1
Lf (k) = Lf (a1,i ). Note that in the previous iteration (qubits a1,i s.t. Lf (a1,i ) = Smax
,
like the wire k being analysed now), all Ef (k)v , for any v 6= k were moved past the
corresponding correction slice and hence Ef (k)k is the first gate acting on the f (k) wire
(as required by both Propositions 6 and 7). Moreover, when the circuit identity in Figure
20-b creates the new CX gates (first step in both RP3 and RP4), it can be pushed
forward to c1,i because the only gate in between is CXkf (k) , which commutes with the
CX gates we want to push forward. Therefore, since all conditions in either Proposition
1
6 or Proposition 7 would be satisfied for all qubits a1,i such that Lf (a1,i ) = Smax
1, the
corresponding RP can be successfully applied.
To complete the proof by induction for the first iteration of the for loop, assume
Algorithm 3 has not aborted in the first q iteration of the first while loop. Then in the
1
(q + 1)th iteration, qubits a1,i such that Lf (a1,i ) = (Smax
q)
1
1
Smin
, where Smin
⌘
mina1,i 2W1 {Lf (a1,i )}, are considered. If a given qubit k is neighbour of a qubit in s(a1,i ),
109
1
then Lemma 10 implies that either k 2
/ W1 or {(k 2 W1 ) ^ [(Smax
q) < Lf (k)} is
true. Here the same analysis as before applies; First, if k 2
/ W1 , Algorithm 3 will apply
RP2, moving the corresponding E gates past slice c1,i . On the other hand, if {(k 2
1
W1 ) ^ [(Smax
q) < Lf (k)}, either the conditions in Proposition 6 or Proposition 7 will
be satisfied, since all E gates acting on qubits in the sets s(a1,v ), for all a1,v 2 W1 s.t.
1
Lf (a1,v ) > (Smax
q), were moved past slice c1,v in previous iterations (with the obvious
exception of gates of the form Evf (v) ). The procedure continues until there is no qubit
left in W1 to be considered. Therefore, Algorithm 3 never aborts when called during the
first iteration (n = 1).
Now let us analyse how the E gates placement changed in the circuit after the nth
iteration of the for loop where Algorithm 3 applied a RP each time it was called by
Algorithm 4. First, note that all E gates acting on qubits in s(i), for i 2 Wn , were moved
to En+1 (via RP2) or transformed into a CX (via RP3 or RP4) and then moved to the
same slice, with the exception of gates of the form Eif (i) which remains in En and will
later be used for the application of the J-gate identity. Moreover, Lemma 9 guarantees
that if an E gate is deleted in a given iteration of the algorithm, it will not be required
in any future step of the algorithm.
Note also that those E and CX gates were moved to the En+1 slice in a specific order,
namely the inverse of the order induced by
f.
Since this ordering is a property of the
associated graph, it does not change during the run of the algorithm. It means that
for any given iteration n of the for loop, the ordering of the two-qubit gates required
for the application of a RP will be in agreement with the order Algorithm 3 will be
called by Algorithm 4. Hence the condition (required for the application of any RP) that
the required two-qubit gates can be pushed trivially to a given slice is always satisfied.
Therefore, at each iteration of the for loop, it arranges all two-qubit gates necessary for
the next iteration in the same order it will be called.
To complete the proof by induction, let us assume that in the first (p
1) iterations of
the for loop Algorithm 3 has not aborted, and then we show that it will not abort in the
pth iteration. Let us analyse the first iteration of the first while loop for n = p. According
p
to Lemma 10, for any ap,i such that Lf (ap,i ) = Smax
, if a given qubit k 2 N (s(ap,i )) then
m
p
either k 2 Wm (if there exists m < p s.t. Smax
> Smax
) or the Jk gate belongs to a layer
Jm0 s.t. m0 > p. If the latter condition is the case, then by definition only RP2 is applied
to ap,i . On the other hand, if k 2 Wm , RP3 or RP4 must be applied. Since by assumption
Algorithm 3 has not aborted in any previous iteration, the conditions for the application
110
of the aforementioned RPs are satisfied and Algorithm 2 will not abort in the current
iteration.
Now assume Algorithm 3 has not aborted in the first q iteration of the first while
loop for n = p. In the (q + 1)th iteration of this loop, qubits ap,i such that Lf (ap,i ) =
p
(Smax
q)
p
p
Smin
, where Smin
⌘ minap,i 2Wp {Lf (ap,i )}, are considered. For a given
k neighbour of a qubit in s(ap,i ), Lemma 10 implies that one of the following statement
m
p
p
holds: (i) k 2 Wm ^[Smax
> (Smax
q)] for some m < p, (ii) (k 2 Wp )^[(Smax
q) < Lf (k)]
or (iii) k 2 Wm0 , for m0 > p. Here the same analysis as before applies; For cases (i) and
(ii) RP3 or RP4 is applied and for (iii) RP2 will move the corresponding E gates past slice
cp,i . Since the same order is respected throughout the algorithm (the inverse of the order
defined by flow), the conditions for the application of the aforementioned RPs are satisfied
and Algorithm 2 will not abort in the current iteration. This procedure is repeated until
there is no more elements in the set Wp , which concludes our proof by induction and
proves the Lemma. ⇤
Theorem 4 Algorithm 4 outputs a compact circuit.
Proof. We show that Algorithm 4 creates |OC | many J-gate blocks (Definition 17) and
then removes |OC | many wires from the circuit (using the J-gate identity), yielding a
compact circuit. The first two conditions in Definition 17 are trivially satisfied for all
SSF extended circuits. In Lemma 11 we proved that Algorithm 4 moves to a future slice
(or removes) all gates E acting on qubits j 2 s(i) (8i 2 OC ) except gate Eif (i) and
hence condition 3 in Definition 17 is also satisfied. Now we prove that condition 4 is
satisfied. Note that all non-input qubits are initialized in state |+i. According to the
SSF construction (Proposition 3), for all qubit j 2 s(i), there exists f
1
(j). By definition
f : OC ! I C and therefore every qubits in s(i), for all i 2 OC , starts in the |+i state.
Therefore, all remaining CXij such that j 6= f (i) can me removed from the circuit without
changing the computation, since CXij |+ij = |+ij (remember that the only gate E acting
on wires j 2 s(i) left behind by the algorithm is Eif (i) ). Therefore, after the step in Line
17, condition 4 in Definition 17 is also satisfied. Since all non-output wires in an extended
circuit are measured in the Z basis by construction, condition 5 is also satisfied and hence
Algorithm 4 has created |OC | many J-blocks after the step in Line 17. Finally, in Line
18 the J-gate identity is applied to all J-blocks, resulting in a compact circuit. ⇤
A step-by-step application of Algorithm 4 to an example is given in Chapter 6.
111
5.3.6
Discussion of results and possible extensions
In the last few sections I have introduced and explored the SSF. I have also developed an algorithm able to obtain a compact circuit from a SSF measurement pattern. In
Chapter 6, I will use the SSF compactification procedure to propose an automated optimization procedure for quantum circuits: starting with a circuit, I will translate it to the
1WQC model as a regflow pattern, optimize it using signal shifting and then translate the
SSF measurement pattern back to the circuit model using Algorithm 4. This automated
optimization scheme explores the global structure of circuit to parallelize several J-gates
and to group together several CX gates, which allows further optimizations to take place
using specific method for Clifford gate parallelization.
Interestingly the scheme fails to compactify the parallel pattern obtained via Pauli
simplification rules [23]. In other words one needs to keep the extra space to keep the
parallel depth obtained due to the Pauli measurements. This further indicates the crucial
role that Clifford computation (corresponding to Pauli measurements) plays in obtaining
the superior parallel power of MBQC over quantum circuits. In the example below, I
explore this time-memory tradeoff when Pauli measurements are considered. I give several circuits (implementing the same computation) from the same measurement pattern,
varying the number of J layers and wires from one compact circuit to another.
Example: Pauli measurements
In this example we explore the role of Pauli measurements in the context of compactification procedures. Let us start by reviewing how the correction operators are modified
when applied to a qubit measured with 0 and ⇡/2 angles (corresponding to Pauli measurements):
⇡
⇡
Mi2 Xis = Mi2 Zis
(5.22)
Mi0 Xis = Mi0
(5.23)
Note that in the MBQC context, both substitutions clearly might reduce the depth of
the computation: Equation 5.22 substitutes Zis for Xis , which can then be removed using
signal-shifting, and Equation 5.23 simply deletes the existing Xis dependency. However,
in the associated extended circuit, these optimisations can not be implemented alongside
the SSF compactification procedure developed in this Chapter (Algorithm 4), forcing one
to choose between optimization in time or memory. To see why, note that Equations 5.22
112
and 5.23 both change the correcting structure for every qubit encoded in the index s. As
a consequence, the step-wise influencing path } (Definition 21), which is the backbone of
the compactification protocol, can not be defined anymore for these qubits.
In summary, if one wants to save memory in the circuit model, the substitutions
in Equations 5.22 and 5.23 must be avoided, allowing the identification of all step-wise
influencing paths and, consequently, the removal of auxiliary qubits. Conversely, if J-gate
parallelization is the goal, Algorithm 4 can be easily adapted to create just |OC |
p many
J-blocks in the extended circuit (instead of |OC | many), where p is the number of Pauli
measurements in the associated measurement pattern.
Consider the circuit shown in Figure 28-a. It has two J gates with arbitrary angles
(arbitrary angles are omitted in the Figure) and two Pauli angles, namely 0 and ⇡2 . Using
the method described in Sec. 4.2.2, we can translate this circuit to the MBQC model; the
associated graph is shown in Figure 28-b and the flow measurement pattern is given by:
⇡/2
Z6s4 Z3s1 X6s5 X3s2 M50 M2 Z2s4 X5s4 M4✓4 Z5s1 Z4s1 X2s1 M1✓1 EG
(5.24)
where, EG = E56 E45 E25 E24 E23 E12 . Using the signal shifting technique (Sec. 5.3.1), we
obtain the following SSF measurement pattern:
⇡/2
Z6s1 +s4 Z3s1 X6s5 +s1 X3s1 +s2 +s4 M50 M2 X2s1 X5s1 +s4 M4✓4 M1✓1 EG
(5.25)
Now let us see what happens when we use the fact that some measurements are Pauli
measurements. Using Equation 5.22, we can optimize the pattern by removing the dependency between measurements 1 and 2, obtaining the following measurement pattern:
⇡/2
Z6s1 +s4 Z3s1 X6s5 +s1 X3s2 +s4 M50 X5s1 +s4 M2 M4✓4 M1✓1 EG
(5.26)
Alternatively, the measurement on qubit 5 can also be parallelised with the ones in 1 and
4, using Equation 5.23. The optimized measurement pattern is the following:
⇡/2
Z6s1 +s4 Z3s1 X6s5 +s1 X3s1 +s2 +s4 M2 X2s1 M50 M4✓4 M1✓1 EG
(5.27)
Finally, both optimizations can be considered together. In this case, all measurements
can be performed at once:
⇡/2
Z6s1 +s4 Z3s1 X6s5 +s1 X3s1 +s2 +s4 M2 M50 M4✓4 M1✓1 EG
(5.28)
113
2
1
1
J
J (pi/2)
4
J
J(0)
4
(a)
1
J
z
J
J(0)
(d)
5
6
1
J
J (pi/2)
4
J
J(0)
(b)
J (pi/2)
4
3
1
J
4
J
(c)
1
J (pi/2)
z
J(0)
J
z
J (pi/2)
4
J
z
J(0)
(e)
(f)
Figure 28: Using compactification procedures to obtain J-gate parallelization for circuits
with Pauli angles. See example 2 for more information.
With a few adaptations6 , Algorithm 4 can be applied to the extended circuits associated to the measurement patterns in equations 5.25 to 5.28. The obtained circuits are
shown in Figures 28-c to 28-f, respectively. Note that the optimization given by equations
5.22 and 5.23 in the MBQC model becomes a J-gate parallelization in the circuit model.
However, it is not clear in which cases the J-gate parallelization implies time optimization. An in-depth analysis of the time-memory tradeoff for extended circuits with Pauli
angles remains as an interesting open question.
5.4
Compact circuits from graphs with generalized flow
In this section, I explore some examples of compactification procedures being applied
to arbitrary generalized flow patterns. The gflows studied here are different from the
ones analyzed in Sec. 5.2 (regflow) and Sec. 5.3 (signal-shifted flow). I start with an
introduction (Sec. 5.4.1) giving some intuition on how compactification procedures are
designed and then I apply it to a couple of examples in Secs. 5.4.2 and 5.4.3. I conclude
this section with a discussion on the applicability of compactification procedures to generic
gflows.
6
Due to the modifications introduced by equations 5.22 and 5.23, it is not possible to create J-blocks
for p many wires, where p is the number of times equations 5.22 or 5.23 were used in the original
measurement pattern.
114
5.4.1
Introduction
By definition, arbitrary gflows may require more than one stabilizer operator to correct
for each measurement. This results in more than one X correction per measurement (as
depicted in Fig. 19) and, consequently, more than one CX gate per cn,i slice. In order to
design compactification procedures for arbitrary gflow extended circuits one must identify
which CX gate corresponds to the one used in the J-gate identity of Fig. 12, and which
CX gates must be removed - it is not obvious how to do this. The identification of this
‘special’ CX gate is also necessary to implement the procedure of removing CZ gates (as
done in Fig. 21), since the procedure reallocates CZ gates to different wires depending on
which CX gate we pick to use the identity on. In what follows I will assume this ‘special’
CX gate has been identified and will be concerned in removing the remaining undesired
CX gates, as necessary for the use of the J-gate identity.
In the examples I present in this section, I remove these unwanted CX gates using
four of the circuit identities in Fig. 20. First, note that Fig. 20-c has a very similar
structure to that of Fig. 20-a, but with CXs instead of CZs, which indicates it may be
helpful in removing CXs in a process similar to the one we described above for CZs.
However, in order to use this circuit identity, another CX gate is required, and it is not
available in the initial entangling round, which consists of CZ gates only. We can make
useful CX gates appear using the circuit identity in Fig. 20-b, which transforms a pair
of CZ gates from the initial entangling stage into one CZ and one CX gate.
The removal of CX gates will follow basically three steps: (1) transformation of a
pair of CZs using the circuit identity in Fig. 20-b, which creates a new CX gate; (2)
moving the CX originated in step (1) forward in the circuit using circuit identity 20-a
when necessary, and (3) applying the circuit transformation in Fig. 20-c, where the leftmost CX is the one generated in step (1), the CX in the middle is the one associated to
the J-gate identity and the right-most is the undesired CX, to be removed by this circuit
transformation.
To clarify the application of this translation method and to see how more compact
circuits are obtained, we now analyze a couple of examples. Note that both examples
are of graphs with gflow, for which the star pattern translation method of [33] fails.
Moreover, those are examples of graphs with no regflow or SSF and, despite this fact, we
obtain compact circuits using our method. The dashed lines in Fig. 29 (first example)
and Fig. 30 (second example) identify the set of gates that will be transformed in each
step using one of the circuit identities of Fig. 20.
115
1
J
1
2
Z
J
3
4
J
4
5
Z
3
Z
4
J
J
(b)
J
Z
2
1
Z
Z
5
5
(a)
1
Z
2
2
3
J
1
(c)
J
Z
2
J
3
4
(d)
J
Z
3
Z
4
5
J
Z
J
Z
1,2
3,4,5
J
J
J
5
(d)
(e)
(f)
Figure 29: Simplifying an extended circuit originated from the entanglement graph with
gflow of Fig. (a). For a step-by-step explanation, see section 5.4.2.
5.4.2
First example
Let us analyze the 1WQC associated with the graph with gflow in Fig. 29-a, where
I = {1, 3} and O = {2, 5}. Since the qubit preparation and entanglement structure is
already defined by the graph, we must describe the measurements and corrections. Qubits
p
1, 3 and 4 are measured onto bases {|±✓i i ⌘ 1/ 2(|0i ± ei✓i |1i)} with respective arbitrary
angles ✓1 , ✓3 and ✓4 . These measurements have correcting sets given by g(1) = {K2 },
g(3) = {K4 } and g(4) = {K2 , K5 }, with stabilizers K2 = X2 Z1 Z3 , K4 = X4 Z3 Z5 and
K5 = X5 Z1 Z3 Z4 . This information completely characterizes the 1WQC, whose associated
extended circuit is shown in Fig. 29-b.
The simplification procedure for this extended circuit goes as follows. In Fig. 29-b we
apply the identity from Fig. 20-b, resulting in the circuit in Fig. 29-c; for this circuit we
need to apply the identity in Fig. 20-a for the dashed box on the right and the adjoint of
the same identity for the dashed box on the left, obtaining thus the circuit in Fig. 29-d.
Now we apply the identities in Fig. 20-a and Fig. 20-c to the dashed boxes on the left
and right, respectively. After the application of these rewrite rules, we end up with the
circuit shown in Fig. 29-e. We can then use the J-gate identity of Fig. 12 to obtain the
final compact circuit in Fig. 29-f.
116
2
1
1
J
Z
2
4
Z
4
6
J
Z
6
Z
2
Z
4
1
J
3
J
6
Z
Z
Z
1,2
J
5
J
Z
4
5
J
(c)
2
J
3
5
(b)
J
Z
6
(a)
1
J
3
4
5
5
Z
2
J
3
3
J
1
J
Z
3,4
J
5,6
J
6
(d)
(e)
(f)
Figure 30: Simplifying the extended circuit originated from the graph in Fig. (a). For a
step-by-step explanation, see Sect. 5.4.3.
5.4.3
Second example
Consider the graph in Fig. 30-a, where vertices 1, 3 and 5 represent measured qubits,
with respective correcting sets g(1) = {K2 }, g(3) = {K2 , K4 , K6 } and g(5) = {K2 , K6 }.
In Fig. 30-b the extended circuit associated with the graph in Fig. 30-a is shown, where
the dashed line encloses the pair of CZs that must be rewritten using the identity of Fig.
20-b. In Fig. 30-c, the dashed line on the left identifies the left-hand side of the circuit
identity of Fig. 20-b, which is the rule applied in this case. Also in Fig. 30-c, we use
the adjoint of the circuit identity in Fig. 20-a to rewrite the set of gates enclosed by the
dashed line on the right. In Fig. 30-d, we again use the circuit identity in Fig. 20-a
to rewrite the gates in the left dashed box and use the rule in Fig. 20-c to rewrite the
sequence of CX gates inside the right dashed box. The same circuit identity in Fig. 20-c
is the one used to reallocate the last two undesired CX gates shown in Fig. 30-e. After
the application of these rewrite rules, we end up with the circuit shown in Fig. 30-f, where
we have already used the J-gate identity of Fig. 12. Note that this final circuit has only
three wires, which is the number of input vertices in the graph of Fig. 30-a.
5.4.4
Discussion
As we pointed out in section 5.4.1, the removal of undesired gates (with the goal of
applying the J-gate identity) relies on the existence of certain CZ gates in the initial
entangling round, as well as the identification of the ‘special’ CX gates, the ones to be
used in the J-gate identities. For instance, in the second example (Sec. 5.4.3), the open
graph under consideration has several different gflows, which means different correction
117
strategies able to perform the same computation. It turns out that for the gflow function
there specified, the special CX gates can be identified in the following way: for every
i, j 2 OC find a k 2 T (i) and a l 2 T (j) (for T (x) = g(x) \ N (x)) such that k 6= l. In
the aforementioned example we have T (1) = 2, T (3) = 4 and T (5) = 6, which means
that the special CX gates are CX12 , CX34 and CX56 . The same identification can be
done for the alternate gflow specified by g(1) = {4, 6}, g(3) = {2, 4, 6} and g(5) = {2, 4},
where the identification T (1) = 6, T (3) = 2 and T (5) = 4 allows the the application of
our method, resulting in a compact circuit. On the other hand, the same is not true for
the gflow specified by g(1) = {4, 6}, g(3) = {2, 4, 6} and g(5) = {2, 6}, since T (1) = T (5).
It remains as an open question to verify if every graph with gflow has at least one gflow
function such that the special CX gates can be identified and a compactification procedure
designed, as it was the case for the examples that we analyzed.
5.5
Concluding remarks
In this chapter I have introduced the concept of compact circuits and the process of
obtaining them from extended circuits, which I have called compactification procedures.
I have developed two algorithms that implement compactification procedures, one for
regflow patterns and the other for SSF patterns. Although the first algorithm gives
circuits which are equivalent to the ones given by SP T , the second gives compact circuits
for a class of measurement patterns which is out of the scope of application of SP T .
Moreover, Algorithm 4 is able to keep the optimized depth of the SSF extended circuit,
since it does not change the structure within the J layers in the SSF extended circuit.
This property of Algorithm 4 will allow us to develop an automated circuit optimization
procedure in Chapter 6. In this automated procedure, we translate a circuit to the
1WQC model, optimize the resulting regflow pattern using the signal shifting rules and
then translate it back to the circuit model as an SSF extended circuit. Finally, we apply
Algorithm 4 to obtain a compact circuit from the SSF extended circuit. Note that such
back-and-forth optimization procedure would not be possible if the translation procedure
used to bring the computation back to the circuit model as a compact circuit did not
preserve the optimization obtained by the application of the signal shifting rules.
118
6
Optimizing quantum circuits
using one-way quantum
computation techniques
In this chapter I give an application of the SSF compactification procedure introduced
and explored in the last chapter, namely the optimization of quantum circuits. In Sec. 6.1,
I describe how a compactification procedure can be used - together with other techniques
- to optimize quantum circuits and give a full example of such a procedure. I conclude
this chapter with a complexity analysis (Sec. 6.2) of the optimization technique proposed
here, comparing it with other known techniques.
6.1
Optimizing quantum circuits by back-and-forth translation
Quantum circuit optimization is a subject of great importance for quantum computing. The enormous technological effort required to perform (even simple) quantum
computing tasks makes the research on circuit optimization a potential shortcut to more
complex quantum computing tasks, bringing quantum algorithms down to the current
technological range. In this chapter I contribute to this discussion by exploring the circuit
optimization that can be achieved by combining several different techniques introduced
and/or explored in this thesis. More specifically, I reduce the number of J layers of a circuit, increasing the number of J gates that can be performed in the same computational
time slice. As I show in Sec. 6.2, this procedure of reducing the number of J layers via
back-and-forth translation results in the reduction of the total number of computational
time slices required to implement a circuit, i.e., the circuit’s depth.
The optimization procedure introduced in this chapter is based on the translation of
a given quantum circuit [38, 86] into a one-way quantum computation [95, 35]. Naturally,
since the two models use different information processing tools, each model has its own
119
optimization techniques. In the 1WQC model, for instance, most of the optimization
techniques are based on the identification of a more efficient correction structure that is
directly linked to the geometry of the underlying global entanglement structure (represented as graphs, as we have seen throughout this thesis). Examples of these techniques
are the signal shifting [35] and the generalized flow [24], which were discussed in Secs.
5.3.1 and 3.3.3, respectively. The so-called standardisation procedure [35] (discussed in
Sec. 3.1.4) can also reduce the number of computational steps by rearranging the 1WQC
operations into a normal form. Moreover, all Pauli measurements in this model can be
performed in the beginning of the computation [96], which is a surprising difference from
the quantum circuit model.
On the other hand, most optimization techniques for quantum circuits are based on
template identification and substitution. For instance, in [49] some circuit identities are
used to modify the teleportation and dense coding protocols, with the purpose of giving a
more intuitive understanding of those protocols. Similarly in [103] and [79] a set of circuit
identities for reducing the number of gates in the circuit for size optimization was given.
In contrast to that, in [84] a useful set of techniques for circuit parallelization was provided, where the number of computational steps is reduced by using additional resources.
However, as noted in [103], all the aforementioned circuit optimization techniques are
basically exchanging a sequence of gates for a different one without any consideration
on the structure of the complete circuit being optimized. The translation into 1WQC
would allow us to explore the global structure of a given circuit, by mapping it to a graph
and analyzing global properties of it, such as flow conditions. The first such scheme by
back and forth translation between the two models was presented in [23]. However the
backward translation into the circuit model required the addition of many ancilla qubits.
Here I show that the addition of ancilla qubits, in order to keep the optimized number of
J layers, can be avoided in some cases by using the compactification procedure introduced
in the last chapter.
The optimization procedure I explore in this chapter, which I will call signal-shifting
circuit optimization (SSCO), can be defined in the following way:
Definition 22 (Signal-shifting circuit optimization) Let C be a circuit composed by
gates taken from the universal gate-set {J(✓), CZ}. The application of the signal-shifting
circuit optimization (SSCO) can be summarized in the following steps:
1. Translate C to the 1WQC model as a regular flow pattern P and associated open
graph (G, I, O) using the technique in Def. 16;
120
2. Apply to P the signal-shifting rules (Eqs. 5.8 to 5.11) until all Zi for i 2 OC have
been removed, resulting in the pattern PSS ;
3. Apply the extended translation technique (Def. 14) to the pattern PSS , generating
the circuit CSS ;
4. Apply the SSF compactification procedure (Algorithm 4) to the circuit CSS , obtaining
the compact circuit CCom as result.
A few comments regarding the choice of the techniques combined in Def. 22 are worth
of note. The first choice, of using star pattern translation to obtain a regflow graph and
pattern from a circuit, is obvious: it is the only known technique able to translate circuits
to the 1WQC model without getting into the trouble of calculating the full unitary being
implemented in one model and then decomposing it appropriately into building blocks
of another model. As discussed in Sec. 5.3, the signal shifting is the best optimization
that can be applied to patterns with regflow (without changing the underlying open
graph) and, therefore, this is the optimization applied in Step 2. To translate the new
correcting structure to the circuit model we use the Extended Translation technique,
which re-interprets each command in a pattern as a quantum circuit element, giving
circuits preserving most of the design of the original pattern: number of qubits, partial
order between operations (gates/commands), etc. In the forth step, in order to preserve
part of the modified structure, specially the number of J layers, we use Algorithm 4 to
remove all qubits added in Step 3, giving a compact circuit as the final product of SSCO
procedure.
In what follows I give a full example of SSCO (Sec. 6.1.1) and then I analyze in
terms of complexity the advantages of SSCO, comparing it with different combinations of
optimization and translation techniques (Sec. 6.2).
6.1.1
Example
In this example I apply the procedure described in Def. 22 to optimize the circuit in
Figure 32-a. The angles of each J-gate are arbitrary and they have been omitted from
the figure. The separation into Steps below is a reference to the four steps in Def. 22.
121
J
J
J
J
J
(a)
In
In/Out In
Out In/Out
In
In/Out
Out In/Out In/Out
Out In/Out
In
In/Out
Out In/Out
In/Out
In
In/Out
In/Out
Out
In/Out
(b)
1
2
3
6
4
5
7
8
(c)
Figure 31: Aplication of the procedure described in Def. 16 to the circuit in Figure 32-a.
See Step 1 of Sec. 6.1.1 for more information.
122
Step 1
Using the method introduced in [23] (summarized in Def. 16 of this thesis), we can
translate the circuit in Figure 32-a to the one-way model, resulting in a open graph
obeying the regflow conditions (Def. 10) and the associated measurement pattern P . The
step-by-step translation to the one-way model is shown in Fig. 31. From the arrows in
Fig. 31-b we can infer the regflow function f :
f (1) = 2
f (2) = 3
f (4) = 5
f (5) = 6
f (7) = 8
With the open graph and function f (i) defined for all i 2 OC , the measurement pattern
can be easily constructed:
X8s7 M7✓7 Z7s5 X6s5 M5✓5 Z7s2 Z5s2 X3s2 M2✓2 Z7s4 Z6s4 Z3s4 Z2s4 X5s4 M4✓4 Z5s1 Z4s1 Z3s1 X2s1 M1✓1
E78 E67 E57 E56 E45 E37 E35 E25 E24 E23 E12
(6.1)
which can by more compactly written using the notation in Eq. 3.1 and the abbreviation
E1
8
⌘ E78 E67 E57 E56 E45 E37 E35 E25 E24 E23 E12 .:
X8s7 X6s5 Z6s4 X3s2 Z3s4 +s1 s5 +s4 +s2 [M7✓7 ]s2 +s1 [M5✓5 ]s4 s4 [M2✓2 ]s1 s1 [M4✓4 ]M1✓1 E1
Step 2
The application of signal-shifting to Eq. 6.2 yields:
8
(6.2)
123
X8s7 X6s5 Z6s4 X3s2 Z3s4 +s1 s5 +s4 +s2 [M7✓7 ]s2 +s1 [M5✓5 ]s4 s4 [M2✓2 ]s1
M1✓1 E1
8
)
Eq.(5.8)
M4✓4 M1✓1 E1
8
)
Eq.(5.11)
✓ 2 s1
✓4
✓1
s4 +s1 [M2 ] M4 M1 E1 8
)
Eq.(5.10)
✓7
s1
✓5 s4 +s1
✓ 2 s1
✓4
✓1
s4 +s1 [M2 ] M4 M1 E1 8
s5 +s4 +s2 [M7 ]S4 s2 +s1 [M5 ]
)
Eq.(5.11)
)
Eq.(5.11)
X8s7 X6s5 Z6s4 X3s2 Z3s4 +s1 s5 +s4 +s2 [M7✓7 ]s2 +s1 [M5✓5 ]s4
✓ 2 s1 s 1
s4 [M2 ] S4
X8s7 X6s5 Z6s4 X3s2 Z3s4 +s1 s5 +s4 +s2 [M7✓7 ] s2 +s1 [M5✓5 ]s4 S4s1
X8s7 X6s5 Z6s4 X3s2 Z3s4 +s1 S4s1
X8s7 X6s5 Z6s4 X3s2 Z3s4 +s1 S4s1
✓4
s1 [M4 ]
✓7
✓5 s4 +s1
✓ 2 s1
✓4
✓1
s5 +s4 +s1 +s2 [M7 ]s2 +s1 [M5 ]
s4 +s1 [M2 ] M4 M1 E1 8
X8s7 X6s5 Z6s4 S4s1 X3s2 Z3s4 s5 +s4 +s1 +s2 [M7✓7 ]s2 +s1 [M5✓5 ]s4 +s1 s4 +s1 [M2✓2 ]s1 M4✓4 M1✓1 E1
8
)
Eq.(5.11)
X8s7 X6s5 Z6s4 +s1 X3s2 Z3s4 s5 +s4 +s1 +s2 [M7✓7 ]s2 +s1 [M5✓5 ]s4 +s1
M4✓4 M1✓1 E1
8
)
Eq.(5.8)
X8s7 X6s5 Z6s4 +s1 X3s2 Z3s4 s5 +s4 +s1 +s2 [M7✓7 ] s2 +s1 [M5✓5 ]s4 +s1 S2s4 +s1 [M2✓2 ]s1 M4✓4 M1✓1 E1
8
)
Eq.(5.11)
)
Eq.(5.8)
X8s7 X6s5 Z6s4 +s1 X3s2 Z3s4
✓ 2 s1
s4 +s1 [M2 ]
✓7
s4 +s1
✓5 s4 +s1
[M2✓2 ]s1 M4✓4 M1✓1 E1 8
s5 +s4 +s1 +s2 [M7 ]S2
s2 +s4 [M5 ]
X8s7 X6s5 Z6s4 +s1 X3s2 S2s4 +s1 Z3s4 s5 +s2 [M7✓7 ]s2 +s4 [M5✓5 ]s4 +s1 [M2✓2 ]s1 M4✓4 M1✓1 E1
8
)
Eq.(5.10)
X8s7 X6s5 Z6s4 +s1 X3s2 +s4 +s1 Z3s4 s5 +s2 [M7✓7 ] s2 +s4 [M5✓5 ]s4 +s1 [M2✓2 ]s1 M4✓4 M1✓1 E1
8
)
Eq.(5.8)
[M5✓5 ]s4 +s1 [M2✓2 ]s1 M4✓4 M1✓1 E1
8
)
Eq.(5.11)
X8s7 X6s5 S5s2 +s4 Z6s4 +s1 X3s2 +s4 +s1 Z3s4 s5 +s4 [M7✓7 ][M5✓5 ]s4 +s1 [M2✓2 ]s1 M4✓4 M1✓1 E1
8
)
Eq.(5.10)
X8s7 X6s5 +s2 +s4 Z6s4 +s1 X3s2 +s4 +s1 Z3s4
[M5✓5 ]s4 +s1 [M2✓2 ]s1 M4✓4 M1✓1 E1
8
)
Eq.(5.9)
X8s7 S7s5 +s4 X6s5 +s2 +s4 Z6s4 +s1 X3s2 +s4 +s1 Z3s4 [M5✓5 ]s4 +s1 [M2✓2 ]s1 M7✓7 M4✓4 M1✓1 E1
8
Eq.(5.10)
X8s7 +s5 +s4 X6s5 +s2 +s4 Z6s4 +s1 X3s2 +s4 +s1 Z3s4 [M5✓5 ]s4 +s1 [M2✓2 ]s1 M7✓7 M4✓4 M1✓1 E1
)
8
X8s7 X6s5 Z6s4 +s1 X3s2 +s4 +s1 Z3s4
✓7
s2 +s4
s5 +s2 [M7 ]S5
✓7
s5 +s4 [M7 ]
Therefore, the SSF pattern PSS is the following:
X8s7 +s5 +s4 X6s5 +s2 +s4 Z6s4 +s1 X3s2 +s4 +s1 Z3s4 [M5✓5 ]s4 +s1 [M2✓2 ]s1 M7✓7 M4✓4 M1✓1 E1 8 .
(6.3)
The difference in depth between the regflow and SSF is shown in the table below:
Method depth partial order
Regflow
5
1
SSF
2
1, 4, 7
f
4
f
s
2
f
5
f
7
2, 5
Step 3
The extended translation (Definition 14) of the measurement pattern in Equation 6.3
gives the circuit in Figure 32-b.
124
Step 4
Now, Let us apply Algorithm 4 to find the compact form of that extended circuit.
We start with the first SSF layer of circuit in Figure 32-b. For n = 1 we have W1 =
1
1
{1, 4, 7}, Smax
= Lf (7) = 5 and Smin
= Lf (1) = 1. Thus, we start with qubit 7. Since
1
N (s(7))\{7} = {;}, no rewrite procedure is applied. The value of Smax
is decreased to 2
(there is no qubit i 2 W1 s.t. Lf (i) = 3 or 4). Now we analyse qubit 4 because Lf (4) = 2.
N (s(4))\{4} = {2, 3, 5, 6, 7}. For qubits in this set it holds that 2
f
5
f
7
f
3, 6. The
maximal value of Lf is stored in N ST EP . For both qubits 3 and 6, Algorithm 4 applies
RP2, moving E65 and E35 past CX45 .
Now N ST EP
N ST EP
1 and the qubit considered is qubit 7. For qubit 7
the algorithm applies RP3, transforming E67 , E57 and E37 into CX68 , CX58 and CX38
respectively and moving those new CXs past the correction structure, removing CX48
from the circuit. Now consider qubit 5 and move E56 and E53 past the correction structure,
using RP2. The same rewrite procedure is applied for qubit 2, moving E23 and E25 past
the correction structure. The rewritten circuit is depicted in Figure 32-c.
1
In the next step, the value of Smax
is decreased to 1, where the qubit to be considered
is qubit 1. N (s(1))\{1} = {2, 3, 4, 5, 6, 7}. For qubits in this set it holds that 4
5
f
7
f
f
2
f
3, 6. For qubit 6, Proposition 4 is satisfied and hence Algorithm 4 applies
RP2, moving E65 past CX15 . Then for k = 3, Algorithm 4 applies RP2 (Proposition
4), moving E35 and E23 past CX12 and CX15 . Next, RP4 is applied to k = 7, moving
CX38 and CX58 past CX13 and CX15 . For k = 5, we have that k 2 s(i) [condition (i)
in Proposition 5] and E35 in slice c1,1 after CX15 gate [condition (ii) in Proposition 5].
Therefore, for this case RP2 moves E35 and E25 past CX12 and CX13 , creating two CZ15
separated by CX15 . By Lemma 8, we know those two CZ15 can simply be removed from
the circuit without changing the computation being implemented. For k = 2, Proposition
4 is satisfied and hence RP2 moves E25 and E23 past CX13 and CX15 . Finally, for qubit
4, RP3 transforms E24 into CX25 and move it past CX12 and CX15 , removing the latter
in the process, resulting in Figure 32-d.
2
2
In the second SSF layer we have n = 2, W2 = {2, 5}, Smax
= Lf (2) = 3 and Smin
=
Lf (5) = 4. Thus, we start with qubit 5. Since N (s(5))\{5} = {7}, Algorithm 4 applies
RP3, moving CX68 past CX56 and CX58 while removing CX58 in the process. After that,
the value of N ST EP decreases and qubit 2 is considered. N (s(2))\{2} = {5, 7} and since
Lf (5) < Lf (7), qubit 7 is considered first. In this case Algorithm 4 applies RP4 moving
gates CX38 and CX68 past the correction structure. Now for qubit 5, RP3 is applied
125
J
J
J
J
1
J
c1,7 c 1,4 c1,1
J
J
2
(a)
c1,7
z
J
z
J
5
J
z
3
4
1
c2,5 c2,2
c 1,4
c 1,1
c 2,5 c2,2
J
2
z
6
z
z
7
J
z
8
3
4
J
z
J
5
(e)
z
6
7
J
z
8
1
J
c1,7 c1,4 c1,1
J
2
(b)
c1,7c 1,4
z
J
z
J
5
J
z
3
4
1
c 2,5 c2,2
c
c 2,5 c2,2
1,1
J
2
z
6
z
z
7
J
z
8
3
4
J
z
J
5
(f)
z
6
7
J
z
8
1
c1,7 c1,4 c 1,1
J
J
2
(c)
J
c1,7 c 1,4 c 1,1
c 2,5 c2,2
J
2
z
z
J
5
J
z
3
4
1
c2,5 c 2,2
z
6
z
z
7
J
z
8
3
4
J
z
J
5
(g)
z
6
7
J
z
8
J
J
J
J
J
(d)
(h)
Figure 32: A complete example of global circuit optimization. In each figure the shaded
gates are the new gates obtained through the RPs (see Section 6.1.1 for more information).
The angles of each J-gate are arbitrary and they have been omitted from the figure.
126
transforming E53 into CX36 . The created CX is moved forward past the correction
structure, removing CX26 from the circuit. The resulting circuit is shown in Figure 32-f.
The command in Line 17 of Algorithm 4 removes the remaining undesired gates using the
trivial identity CXij |+ij = |+ij . The resulting circuit is depicted in Figure 32-g. Finally,
the J-gate identity is applied for every qubit i 2 OC resulting in the optimised compact
circuit shown in Figure 32-h.
6.2
Complexity analysis and discussion
In this section I analyze the space and depth complexity of the SSCO procedure introduced in this chapter. In special, I show that this method gives circuits less demanding in
both time and space, if compared with the general method described in [23]. An overview
of this analysis is shown in Figure 33.
Let us start by counting the number of wires and the depth of the computation in the
first three steps of SSCO (Def. 22), since they consist in previously analyzed optimization
scenarios. Let C be the initial quantum circuit with n wires, m = poly(n) J-gates and
depth d = poly(n). We can translate the computation implemented by C to a regflow
pattern P with m + n qubits, out of which m are measured qubits, and depth dP  d [23].
This pattern can be further optimized to depth dSS  dP by performing signal shifting
[23] and obtaining a new pattern PSS . The extended quantum circuit CSS , corresponding
to the signal shifted pattern PSS , can be created via the method given in Definition 14.
This new circuit CSS performs exactly the same computation as the initial circuit C and
has the same number of J-gates, m + n wires and depth dSS · O(m). One could apply the
parallelization method of [84], which states that a series of k controlled gates connecting
the same input to k target qubits can be parallelized to O[log(k)] depth using O(k)
ancillas, to create a parallelized circuit CPar with depth dSS · O(log n) and size O(m2 ) [23].
Depending on the value of dSS , the depth of CPar can be smaller than that of the original
circuit C, but this could increase the space used, which can increase considerably, i.e from
n to O(m2 ).
To resolve this problem, instead of parallelizing the circuit CSS we apply the compactification procedure (Algorithm 4). This will give us the compact circuit CCom (Step 4 in
Def. 22) with n wires and dSS · O(deg(G)) where deg(G) is the degree of the open graph
(G, I, O) of PSS .
Lemma 12 (Lemma 28 in [42]) Let CSS be the quantum circuit obtained through
127
measured
qubits
space
depth
#J gates
Initial quantum circuit (C)
n
d = poly(n)
m = poly(n)
Regflow pattern
n+m
dP  d
m
n+m
dSS  dP  d
m
SSF extended circuit (CSS )
n+m
dSS · O(m)
Parallelized circuit (CP ar )
O(m2 )
dSS · O(log m) = dSS · O(log n)
m
Compact circuit (CCom )
n
dSS · deg(G)
m
Parallelized compact
O(n2 )
SSF pattern
circuit (CComP ar )
dSS · O(log(deg(G))) = dSS · O(log n)
m
m
Figure 33: A summary of the optimization procedure using extended translation and
compactification. Each arrow on the left of the table indicates a different optimization
or translation procedure. The parallelization procedure described in [23] (without Pauli
simplification) ends with the “Parallelized Circuit” whereas our optimization can end with
a “Compactified Circuit” or a “Parallelized Compactified Circuit” depending the goal of
the optimization.
extended translation from a signal shifted measurement pattern PSS on an open graph
(G, I, O) with depth dSS . The application of the SSF compactification procedure (Algorithm 4) to CSS results in a circuit with depth dSS · O(deg(G)).
Proof.
Algorithm 4 applies Rewrite Procedures (2) - (4) (as defined in Section 5.3.4)
to the extended circuit of PSS . Note that the number of J-gate layers J in CSS is equal to
the depth of PSS [23]. Since the rewrite procedures do not change the J-gate layers, the
compactified circuit CCom will also have dSS J-gate layers.
Now we will count the number of gates in-between the J-gates. We do not need to be
concerned with gates originally in the C layers, since Algorithm 4 removes all of them. As
all of the gates in E layers correspond to edges in the graph G, the maximum number of
such gates has to be O(deg(G)). According to the rewrite procedures, none of the CXk,i
gates created will be moved past the J-gates. As can be seen from the rewrite procedures
and Algorithm 4, these will be created because of the existence of some Ef
CSS . Since these correspond to the edges connected to the vertex f
1
1 (i)k
gates in
(i) there can only
exist O(deg(G)) many CX gates between any two J-gates. Now we know that there can be
a total of O(deg(G)) two qubit gates in-between any two J-gates and hence also between
the J-gate layers. Therefore the total depth of the circuit CCom will be dSS · O(deg(G)).
⇤
128
Note that if deg(G) < log n the CCom will have smaller depth than CSS while the
corresponding space will be considerably smaller. We can further decrease the depth of
CCom by applying the parallelization method from [84]. The depth of the new circuit
CComPar would be dSS · O(log(degG)) and size O(n2 ). Moreover, due to the way C is
translated into P (that is, using the method described in 16), the maximum number of
edges connected to a vertex in G will be O(d), where d = poly(n). Therefore the depth of
CComPar can be written as dSS · O(log n). Hence the compactification procedure together
with the parallelization method from [84] will in the worst case give us the same depth as
the method from [23], but uses considerably less qubits [n2 vs m2 , where m = poly(n)].
To conclude this chapter, I would like to comment on a possible simplification of the
procedure in Def. 22. The optimization procedure that I have introduced in this chapter
is able to reduce the depth of a circuit by reducing the number of J layers in it, via
back-and-forth translation to the 1WQC model. It is clear, however, that the optimized
circuit can be obtained from the original one directly in the circuit model; since both are
elements from the same model, there might exist a direct procedure that takes one to
another and vice-and-versa. What is not clear is whether a direct procedure within the
circuit model would be easier to implement than the one introduced here.
We can learn a couple of things from the example in Sec. 6.1.1. Note, for instance,
that the number of two-qubit gates in the original and optimized circuits is preserved,
although they might change from CZ to CX gates in some cases. The number of J
gates is also preserved, as they are just reallocated in the circuit. I conjecture that the
same optimization could be implemented within the circuit model using just the following
identity and a prescription of which qubits and in which order the identity must be applied:
Jj (✓)CZij = CXij Jj (✓).
(6.4)
This identity can be obtained by noting that J(✓) = HP (✓) and Hj CZij Hj = CXij . It
is easy to check by trial and error that this identity is sufficient to optimize the simple
example analyzed in Sec. 6.1.1, but in order to prove that it holds for every circuit
optimization obtained by Def. 22, more work should be done.
Therefore, I leave the existence of a simple procedure within the circuit model that is
able to implement the same optimization provided by the method in Def. 22 as an open
question. It is important to note, however, that independently of there being a simple
procedure able to give the same optimized circuits as Def. 22, what I have shown here is
that our circuit optimization is the best that can be done by using flow-like techniques,
129
since all circuits can be mapped to the 1WQC model as graphs with regflow and SSF is
the optimal flow for such graphs.
130
7
Closed timelike curves in one-way
quantum computation
The possibility of time travel has been studied for decades in the context of general
relativity. Several decades after Einstein developed the general theory of relativity, the
first explicit spacetime geometry containing closed timelike curves (CTCs) - paths in
spacetime that would allow a physical system to interact with its former self - was proposed
by Kurt Gödel [50]. After Gödel’s work, a variety of spacetimes containing CTCs were
proposed [18, 51]; in fact, it was later realized that CTCs are a generic feature of highly
curved, rotating spacetimes [18, 109, 71]. Nevertheless, it is not clear closed timelike
curves can in fact exist: Stephen Hawking, for instance, suggested that the conditions
necessary to create CTCs could never exist in any physically realizable spacetime - a
concept known as Hawking’s chronology protection postulate [59]. For instance, it was
shown that cosmic string geometries can contain CTCs [51] but cannot create them from
scratch [37, 31].
Assuming that closed timelike curves exist, a series of results were obtained regarding
their implications for quantum mechanics and quantum computation [39, 12, 28, 15, 92,
93]. In this chapter I describe how the one-way model of measurement-based quantum
computation [95] encompasses in a natural way a model for CTCs proposed by Bennett
and Schumacher [1], and more recently (and independently) by Svetlichny [106]. I show
that the one-way model effectively simulates deterministically a class of CTCs in this
model, and characterize this class. A second model for CTCs is Deutsch’s highly influential
study of quantum time-travel [39]. I show that Deutsch’s model leads to predictions
conflicting with those of the one-way model, and identify the reason behind this.
This chapter is organized as follows. In Sec. 7.1 I give a brief summary of the recent
revival of interest on CTCs by the quantum information community, highlighting the
main contributions over the last years. Then, in Sec. 7.2, I review the CTC model
based on quantum teleportation and postselection proposed by Bennett, Schumacher and
131
Svetlichny. Then, in Sec. 7.3 I discuss how CTCs appear naturally in the one-way model,
and show that they correspond to CTCs in the Bennett/Schumacher/Svetlichny model. In
Sec. 7.4 I characterize a class of CTC-assisted quantum circuits that can be rewritten as
time-respecting circuits (i.e., with no CTCs). I give an explicit (although not completely
general) method for verifying if a CTC-assisted circuit can be simulated using the 1WQC
model. Next, in Sec. 7.5, I review the highly influential CTC model proposed in 1991
by David Deutsch [39]; I use Deutsch’s consistency conditions to calculate the predictions
for the same CTC-assisted circuits analyzed in Sec. 7.2 using the BSS model. I show
that while the BSS model encompasses the one-way model prediction, Deutsch’s model
gives completely different predictions. Finally, in Sec. 7.6 I discuss a recent proposal
by Raussendorf et al. [99] for using 1WQC as a quantum mechanical toy model for
space-time.
7.1
Review of the literature
In 1991, David Deutsch published his studies on how quantum mechanics would behave in the presence of closed timelike curves (CTCs) [39]. In this seminal paper, Deutsch
argues that by treating CTCs, and the particles interacting with it, as quantum mechanical objects, some paradoxical scenarios obtained using classical physics can be avoided.
Moreover, it would allow one to “clone” a quantum system (which is prohibited in quantum theory - a result known as the no-cloning theorem) and to distinguish between nonorthogonal states. Approximately one decade later, several papers analyzing CTCs from
the quantum information and computational complexity perspectives started popping up.
In general, those papers address the following question: what is the computational power
of a quantum computer if it is allowed to interact with closed timelike curves? In 2002,
Todd Brun [27] explicitly showed how CTCs could be used by a classical computer to
factor a number into its prime factors; he also argued that the same mechanism could
be used to show that classical computers assisted by Deutschian CTCs1 would be able
to solve many hard problems in polynomial time, as NP-complete and PSPACE-complete
problems2 . In 2003, David Bacon analyzed [12] the computational power of quantum
computers assisted by Deutschian CTCs, concluding that they would be able to efficiently solve NP-complete problems with only a polynomial number of quantum gates.
A few years later, in 2008, S. Aaronson and J. Watrous showed [7] that, if CTCs exist
1
That is, CTCs modeled as suggested by David Deutsch in [39].
For more complete descriptions of the computational complexity classes that will be used in this
thesis, please see: https://complexityzoo.uwaterloo.ca .
2
132
and can be described using the Deutsch model, quantum computers would be no more
powerful than classical computers. In fact, both types of computers would be extremely
powerful, being able to solve PSPACE-complete problems.
In 2009, Brun et al. [28] suggested that the assistance of Deutschian CTCs would allow
perfect discrimination between any set of quantum states. For instance, they give a CTCassisted circuit that would allow one to distinguish the four BB84 states {|0i, |1i, |+i, | i}.
In the same year, Bennett et al. [15] contested the claims in [28], pointing out that the
apparent power of Deutschian CTCs comes from analyzing the evolution of pure states
and then extending those results linearly to mixed states. It is not clear, however, that
the very concept of mixed states would be well-defined in such CTC-assisted scenarios.
More specifically, Bennett et al. point out an odd consequence of the results in [28],
namely the discrimination of identical states: in [28], they showed how would be possible
to map the linear dependent set of states {|0i, |1i, |+i, | i} to four orthogonal states by
using the power of Deutschian CTCs, and in [15] the authors point out that it would have
as a consequence the ability to discriminate the identical mixed states I/2 =
|1ih1| and I/2 =
1
2
1
2
|0ih0| +
|+ih+| + | ih | . An active discussion on the ability of CTC-
assisted quantum circuits to distinguish non-orthogonal states, clone quantum states and
to perform other post-quantum-theory tasks has been taking place since then [29, 30, 93,
70, 102, 9, 111].
In 2009, George Svetlichny suggested a new way of analyzing CTC-assisted quantum
circuits [106]: instead of requiring Deutsch consistency conditions, he proposes the use of
an adaptation of the quantum teleportation protocol where one is allowed to post-select
measurement outcomes. In fact, this model had been independently proposed by Bennett
and Schumacher in an unpublished joint work [1] presented at the quantum information
processing school QP IP (Mumbai, India), in February 2002. Therefore, from this point
on I will refer to this CTC model as the BSS model, after Bennett, Schumacher and
Svetlichny.
The BSS model leads to less powerful CTC-assisted circuits than the Deutsch model
does. A BSS circuit would be able to solve problems in the complexity class post-BQP
(which is believed to be larger that NP but properly contained in PSPACE), and since it was
proved in [5] that post-BQP = PP, we say that BSS circuits are able to solve PP-complete
problems. Therefore, BSS CTCs would be less powerful than Deutschian CTCs, which
apparently can solve PSPACE-complete problems.
In 2010, in collaboration with Ernesto F. Galvão and Elham Kashefi, I proposed a
133
framework to analyze CTCs in the 1WQC model [41]. We showed that 1WQC algorithms
can be formally manipulated (using the stabilizer formalism) in order to allow the appearance of anachronical operators, which become anachronical gates when translated to
the circuit model. Those circuits with anachronical gates can be further manipulated
(simply by adding a SW AP gate and re-drawing the circuit) to become CTC-assisted
circuits. We also showed that those CTC-assisted circuits originated from anachronical
1WQC algorithms encompass the results given by the BSS model, while showing a general disagreement with the predictions of the model proposed by Deutsch. This chapter
is mainly based in the results of that paper.
In the same year an experimental simulation [76] and a deeper theoretical analysis [75]
of the BSS model was uploaded to the preprint repository Arxiv.org by Lloyd et al., being
published a few months later. Those papers reinforced our result that the BSS model
seems to be more adequate to describe CTC-assisted circuits than the model proposed by
Deutsch.
The research on the power of CTCs seems to be far from an end, since there is still
no agreement on which of the existing CTC models (BSS and Deutsch’s) should be used
to describe CTC-assisted circuits. In fact, there is still no good reason to choose one
of those two models and maybe some new, interesting ways of analyzing CTC-assisted
circuits may be proposed in the future.
Remark on terminology. Contrary to most of the papers that analyze CTCs
using the model proposed by C. Bennett and B. Schumacher [1] and independently by
G. Svetlichny [106], I use the expressions “BSS CTCs” and “BSS model” (in reference to
Bennett/Schumacher/Svetlichny - BSS) as opposed to “P-CTCs” and “P-CTC model” (in
reference to postselected CTCs) in order to give due credit to the authors of the model,
as is done with the Deutsch model for CTCs [39].
7.2
A model for CTCs based on teleportation and postselection: The BSS model
In this section I review the main features of the CTC model proposed by Bennett and
Schumacher [1] and by Svetlichny [106], the BSS model. This model combines quantum
teleportation with post-selection in an attempt to predict the behavior of quantum mechanics near closed timel-like curves. Since this model relies on post-selection, anything
that happens in a spacetime with BSS CTCs can also happen in conventional quantum
134
B
C
V
B'
|β 〉
00
C'
B
C
(a)
V
B' 〈β00|
C'
(b)
Figure 34: a) CTC takes qubit back in time to interact with its past self; b) Bennett/Schumacher/Svetlichny (BSS) circuit to simulate this CTC probabilistically, using
teleportation and post-selection (see main text).
mechanics with some finite probability.
Ideas similar to the BSS model were proposed independently in several different contexts. For instance, the combination of entanglement and deterministic projections (instead of quantum measurements) were studied in the context of black hole evaporation
by Horowitz and Maldacena [61] and Gottesman and Preskill later noted that this scheme
could be used for time travel [53]. Pegg explored a similar mechanism for probabilistic
time machines [90]. Chiribella et al. also used this mechanism when studying extensions
of the quantum computational model [32] and Brukner et al. when analyzing probabilistic teleportation, where only desirable results were retained, as a computational resource
[26].
More recently in [75], Lloyd et al. discussed several aspects of the BSS model, as for
instance its consistency with path integral approaches for spacetimes containing closed
timelike curves. The first experimental implementation of the BSS model was published
in [76], where the authors implemented a simulation of the grandfather paradox.
For simplicity, I will restrict our discussion to two-qubit unitaries; the generalization
to larger-dimensional systems is straightforward. In Fig. 34-a I represent a circuit with a
CTC that takes the top qubit back in time to interact with its past self via the two-qubit
unitary V . This CTC is simulated using teleportation in BSS’s construction (see Fig. 34p
b). Two qubits are prepared in the Bell state | 00 i = 1/ 2(|00i + |11i), with one of them
sent though V together with an arbitrary input state |
in i
at position B in Fig. 34-b.
After V we perform a Bell-state measurement, postselecting those events corresponding
to projection onto the initial |
00 i.
The post-selected teleportation guarantees that the
state at C is state B 0 teleported back in time to interact with state B via V . The scheme
works only probabilistically, implementing a map from state B ! C 0 .
In the absence of yet-undiscovered physical CTCs, quantum circuits such as the BSS
135
|+〉
|ψ〉
in
|+ 〉
〈+|
V
〈+|
〉
|ψout
Figure 35: Rewritten BSS circuit using preparation of and projections onto |+i states.
The unitary V is decomposed using the universal gate-set consisting of J(✓) and CZ.
|+〉
|ψ〉
in
|+ 〉
〈+|
J-Θ
〈+|
〉
|ψout
V
Figure 36: BSS circuit simulating a CTC with unitary V = [J( ✓) ⌦ I]CZ.
circuit in Fig. 34-b simulate the CTCs with a finite probability of success. Svetlichny’s
model differs from that of Bennett and Schumacher in an irrelevant detail only: the
unitaries he considered involved a swap between the states to be fed to V , that is, he
modelled CTCs such as the one in Fig. 34-a with unitaries of the form V = U · SW AP .
In order to link the BSS CTC simulation circuits with the one-way model of quantum
computation [95] we consider the universal gate set {J(✓), CZ} [34]. We can rewrite any
p
BSS circuit using these gates, initialization in state |+i ⌘ 1/ 2(|0i + |1i), and final Pauli
X measurements that post-select projections onto state |+i (see Fig. 35). This is closely
related to the setting of the one-way model, so we can use various tools developed in that
context to study CTCs as described by the BSS model.
Let us work out the BSS simulation for one particular CTC of interest, given by
unitary V = [J( ✓) ⌦ I]CZ. The BSS circuit that simulates this particular CTC is
shown in Fig. 36. The circuit acts on input state |
|+i with probability |↵ + e
i✓
in i
= ↵|0i + |1i to output state
|2 /4, as can be easily checked. The nonzero probability of
projecting onto |+i state means the BSS formalism predicts that the action of this CTC
is to deterministically project the input state onto |+i.
Interestingly, the finite probability of success is the mechanism that avoids paradoxical
scenarios to occur, i.e., situations in which the combination of input state and interaction
V prevents the existence of a self-consistent state for the time-traveling system. In such
scenarios, the BSS model yields a probability of success equal to zero, as noted in [1, 75].
136
This mechanism encompasses in a natural way the Novikov principle [48], which states that
only logically self-consistent events can occur in the Universe. Note also that any quantum
measurement yields the same statistical results (and here I include correlations with timerespecting qubits) independently of whether it is performed in the system entering the
BSS CTC or exiting it, since the BSS simulation is constructed by projecting out part of
a pure state. Contrary to Deutschian CTCs, which typically take pure states to mixed
states, BSS CTCs take pure states to pure states. Therefore, BSS CTCs behave like an
ideal quantum channel, taking quantum states to the past [58].
7.3
Closed timelike curves in one-way quantum computation
In this section I show how CTCs appear naturally in the one-way model of quantum computation [96]. The key element is the appearance of anachronical dependencies
in measurement patterns, obtained using the stabilizer formalism. This property was explored in depth in Chapter 3, where it was shown that deterministic measurement patterns
[associated to an open graph (G, I, O)] can be constructed in two steps: (i) start with
a projection-based pattern, where every qubit i 2 OC is projected onto state |+✓i i [Eq.
(3.40)] and (ii) look for a collection of stabilizer operators Kj (j 2 I C ) that removes every
anachronical Z from the pattern without creating new ones. In the cases step (ii) can
be successfully implemented, the new pattern is a physically sound measurement pattern,
with no projective operations. Here we study these problematic time dependencies in
more detail, and show that they correspond to BSS CTCs when translated to the circuit
model.
For concreteness, let us start by analyzing a simple pattern implementing a one-qubit
unitary:
X2s1 M1✓ CZ12 N2 |
in i1 .
(7.1)
This measurement pattern was analyzed before in this thesis in Chapters 3 and 5, but for
the sake of convenience let us review again what it is doing. This sequence of operations
can be represented as a two-qubit quantum circuit, see Fig. 37-a. An arbitrary input
state |
in i1 ,
previously entangled via a CZ gate with a qubit initially in state |+i2 , is
then measured in the |±✓ i basis. The outcome s1 = 0 or 1 controls classically whether or
not to apply a Pauli X gate on qubit 2. Fig. 37-b represents the same operations, only
with a Z basis measurement and with the controlled operation implemented coherently
137
|ψ〉
|+ 〉
|ψ〉
|+ 〉
MΘ
X
(a)
J-Θ
MZ
(b)
Figure 37: Two equivalent circuits, i.e. implementing the same unitary. In a) we have
a classically-controlled X unitary dependent on the measurement outcome of |±✓ i basis
projection. In b) this has been turned into a coherent circuit with measurement onto the
Z basis.
J-Θ
|ψ〉
in
|+ 〉
J-Θ
(a)
|ψ〉
out
|ψ〉
in
|+〉
(b)
Figure 38: a) Circuit that includes a CTC with an anachronical CZ gate; it is equivalent
to the two circuits in Fig. 37. b) The same circuit rewritten in the BSS format.
as a CX gate. The two circuits are equivalent as they implement the same unitary J( ✓)
on initial state |
in i1 ,
with output in qubit 2.
It is easy to find other patterns (and corresponding circuits) which are equivalent to
sequence (7.1), i.e., implement the same unitary J( ✓) between input and output qubits.
We start by observing that the state to be measured |Gi = CZ12 |
in i1 |+i2
is stabilized
by the two operators {1 = Z10 X20 , Z11 X21 }, as Z1 X2 |Gi = |Gi. In other words, Z1s1 X2s1 is
a stabilizer of |Gi independently of whether s1 = 0 or 1. This enables us to manipulate
sequence (7.1) as follows:
X2s1 M1✓ |Gi = X2s1 M1✓ Z1s1 X2s1 |Gi = M1✓ Z1s1 |Gi.
(7.2)
This last sequence represents a time-travel conundrum: a classically controlled Pauli Z
unitary which must be applied depending on the outcome of an as-yet unmeasured qubit.
This is turned into a quantum CTC if we apply the anachronical Pauli Z operation
coherently, as we see in Fig. 38-a. A rewriting of this circuit in slightly different form
(Fig. 38-b) shows that the top qubit enters exactly the CTC we analyzed using the BSS
model (see Fig. 36). We can now compare the predictions of the one-way model for this
particular CTC with those given by the BSS model.
138
|+〉 1
|+〉 2
3
|ψ〉
in
|+〉 4
〈+|
J-Θ
〈+|
Figure 39: Circuit that implements the probabilistic BSS simulation of the CTC circuit
in Fig. 38-b.
An apparent mismatch between the BSS formalism and the one-way model appears
when we analyze the action of the CTC in Fig. 38-b. On the one hand, our analysis
of the circuit in Fig. 36 has shown that the CTCs effect is to project the input state
onto |+i. On the other hand, comparison between Figs. 37-a and 38-a suggests that the
effect of the CTC should be to project the top qubit onto |+✓ i instead. This is because
a post-selected |+✓ i1 projection in the circuit of Fig. 37-a is what it takes to implement
the unitary J( ✓) to |
in i,
without the need for the controlled X correction.
The resolution of this apparent conflict is surprising. The one-way model only predicts
that, when embedded in the circuit of Fig. 38-a, the CTC should implement the same
input-output map J( ✓) as its two equivalent circuits in Fig. 37. Using the circuit in Fig.
36 to simulate what happens in the CTC of Fig. 38 is unwarranted; instead, we should
simulate the CTC’s action when embedded in the circuit of Fig. 38. The BSS circuit for
this simulation is in Fig. 39. A simple calculation shows that this circuit, at once, fulfills
the predictions of both the BSS and the one-way model: it projects qubits 3 and 4 onto
state |+i3 ⌦ [J( ✓)|
in i]4 .
This illustrates what seems to be a general feature of CTCs: their effect extends not
only to the time-travelling sub-system A, but to all sub-systems that have interacted
with A prior to A’s encounter with CTCs. In the context of the one-way model what
interests us is the dynamics the measured (time-travelling) qubits induce on the output
(time-respecting) qubits. It is this dynamical map that we can calculate and compare,
as we have done in this section. This will also be the key that allows for the comparison
with Deutsch’s CTC model in Sec. 7.5.
139
7.3.1
Obtaining CTC-assisted circuits from one-way patterns
The simple form of the one-way pattern in Eq. (7.1) may suggest that the CTCs
that appear in the one-way model are only as simple as the one appearing in Fig. 38,
with CTC unitary V consisting of only two gates, a CZ and a J(✓). In fact, a given
deterministic one-way computation can be equivalent to the simulation of different CTCs
implementing the same input-output map, and these CTCs may have different structures.
To illustrate this, let us consider the following sequence of commands implementing a oneway computation:
(7.3)
X2s4 X1s4 Z1s3 M4✓4 X4s3 M3✓3 |Gi,
where
|Gi = CZ23 CZ13 CZ14 CZ34 N4 N2 N1 |
in i3
(7.4)
is the state associated with the graph in Fig. 40-a. The extended translation of the
measurement pattern in Eq. (7.3) is shown in Fig. 40-b. Note that as the sequence in
(7.3) is time-respecting, so is the associated circuit.
We can now obtain different sequences of one-way operations that implement the same
map, by rewriting the initial state |Gi as Ki |Gi, with Ki being a stabilizer of |Gi. It is
easy to check that the following operators are stabilizers of |Gi:
K1s4 = X1s4 Z3s4 Z4s4
(7.5)
K2s4 = X2s4 Z3s4
(7.6)
K4s3 = X4s3 Z3s3 Z1s3
(7.7)
Note that as qubit 3 is in an arbitrary input state, K3 = X3 Z1 Z2 Z4 is not a stabilizer of
|Gi. Using stabilizers K1 , K2 , K4 we obtain three new sequences of operations that now
include anachronical corrections. For example, after applying K2s4 we have the following
pattern:
X1s4 Z1s3 M4✓4 X4s3 M3✓3 Z3s4 |Gi,
(7.8)
which has the anachronical command Z3s4 . A different anachronical pattern can be obtained by using the identity K4s3 |Gi = |Gi in Eq. (7.3):
X2s4 X1s4 M4✓4 M3✓3 Z3s3 |Gi,
(7.9)
which has the anachronical command Z3s3 . Again, we can obtain a new anachronical
140
1
2
3
4
1
3
J
z
J
4
2
(a)
z
(b)
1
1
2
3
2
3
J
z
J
4
J
z
4
z
J
z
(c)
(d)
1
1
2
2
3
J
z
4
J
z
3
J
z
4
J
z
(e)
(f)
1
1
2
2
3
J
J
4
(g)
z
3
z
4
J
z
J
z
(h)
Figure 40: (a) Entanglement graph corresponding to state |Gi in Eq. (7.4). On this
state we can perform the sequence of one-way operations in Eq. (7.3). The extended
translation of Eqs. (7.8), (7.9) and (7.10) are shown in (c), (e) and (g), respectively. In
Figs. (d), (f) and (h) I redraw the CTC circuits in Figs. (c), (e) and (g), respectively, so
as to explicitly show the CTC as a time-travelling qubit.
141
pattern by using the identity K1s4 |Gi = |Gi in Eq. (7.3):
X2s4 Z1s3 M4✓4 Z4s4 X4s3 M3✓3 Z3s4 |Gi,
(7.10)
which has the anachronical commands Z3s4 and Z4s4 .
Each such sequence can be translated into the circuit model, where the classically
controlled X and Z corrections appear as coherent CX and CZ gates. These anachronical
gates correspond to CTCs when translated to the circuit model. The extended translation
of Eqs. (7.8), (7.9) and (7.10) are shown in Figs. 40-c, 40-e and 40-g, respectively. In Figs.
40-d, 40-f and 40-h I redraw the CTC circuits in Figs. 40-c, 40-e and 40-g, respectively,
so as to explicitly show the CTC as a time-traveler qubit.
Any deterministic one-way pattern yields a class of CTC circuits simulatable by it.
These circuits are obtained as shown above, by using arbitrary stabilizers that introduce the anachronical dependencies in the (originally deterministic and time-respecting)
pattern.
7.4
Deterministic simulations of CTCs
Note that a deterministic simulation of the BSS circuit in Fig. 39 (with respect to its
action on input state |
in i)
is achieved by the circuit in Fig. 37-b, which effectively imple-
ments unitary J( ✓). In other words, the circuit in Fig. 37-b simulates deterministically
the CTC circuit in Fig. 38-b. A natural question is then to determine which BSS CTCs
can be simulated deterministically by a one-way pattern and its equivalent circuit. In this
section I present a systematic way to find measurement patterns that deterministically
simulate CTCs in BSS model.
We start with the BSS circuit in which each CTC is simulated by preparation of state
|
00 i
and subsequent postselected projection onto the same state. The first step is to
translate the BSS circuit into a one-way projection-based pattern3 . We translate the BSS
circuits using a simple variation of the star pattern translation technique (introduced in
[23] and summarized in Def. 16, Chapter 4). This variation consists in the modification
of Steps 1 and 3 and the addition of another Step in Def. 16. First, the technique in Def.
16 does not account for postselection, necessary to translate the post-selected Bell-pair
measurements in the BSS circuit. A postselected projection is translated to the one-way
3
|✓i i
That is, a pattern with projections Pi
corrections, as in Eq. (3.41).
instead of measurements Mi✓i and time-respecting Pauli
142
|+i
model as expected, that is, as a vertex i being acted upon by the operator Pi
= Mi0 Zisi
(a post-selected Pauli X measurement). Moreover, a new subcase shall be added to
Step 3 in Def. 16 in order to take into account the new label that refers to postselected
measurements. Next, we need to modify Step 1 in Def. 16 in order to give a projectionbased pattern implementing the J(✓) gate. We do this translation by replacing each J
gate in the CTC-assisted circuit by a horizontal edge connecting to vertices. Label the
vertex on the left as “input” and the one in the right as “output”. In the associated pattern,
the qubit in the left is projected onto state |+✓ i. These modifications are summarized
in Def. 23, where I give a complete description of the method that allows one to find
time-respecting circuits implementing the same map as some CTC-assisted circuits.
The resulting pattern implements the same input-output map as the original BSS circuit and it can be manipulated using stabilizer operations (such as local complementation
[107]), with the aim of eliminating the ancillas added in the |
00 i
state preparations and
postselections required by the BSS simulation circuit. This can always be done since any
such Bell-pair projections translate only as a sequence of Pauli projections, which enables
us to apply the general rules for removing a Pauli measurement from a measurement
pattern [60]. In some cases this results in a pattern where the anachronical corrections
(added during the translation of the J gate) can no longer be eliminated, resulting in
anachronical circuits corresponding to unsound physical operations.
The next step is to turn the projection-based pattern into a runnable measurement
pattern (Def. 5). In some cases it will not be possible to do so, which is reasonable
since CTC-assisted quantum circuits are believed to be more powerful that regular quantum circuits, as will be discussed in Sec. 7.5.3. In other cases, however, the open graph
associated to the resulting pattern satisfies the determinism conditions for the one-way
model (the flow conditions) discussed extensively in Chapter 3. If that is the case, the
anachronical corrections that appear can be removed. Note that the operations used in removing the ancillas introduced by the BSS simulations consist of Local Complementation,
removal of Pauli measurements, and other stabilizer manipulations, all of which preserve
the map implemented from input to output. As a result, the newly-found sequence (and
its equivalent circuit) implements deterministically the same map that succeeded only
probabilistically in the BSS simulation circuit. This effectively characterizes a class of
BSS CTC circuits that admit a deterministic simulation in the one-way model. The
method of obtaining a time-respecting circuit from a CTC-assisted circuit via one-way
model (when it is possible) is summarized in Def. 23.
143
Definition 23 (Obtaining time-respecting circuits from CTC-assisted ones) Let
Cctc be a CTC-assisted circuit composed by gates from the gate-set {J(✓), CZ}. Then, a
time-respecting circuit performing the same input to output map can be obtained if all
steps below can be accomplished succesfully:
1. Apply the method for obtaining patterns from circuits described in Def. 16 with the
following modifications:
• (Step 0) A postselected projection is translated as a vertex i with label “post|+i
s” being acted upon by the operator Pi
= Mi0 Zisi (a post-selected Pauli X
measurement);
• (Step 1 modified) Replace each J-gate by a horizontal edge connecting two vertices. Label the vertex on the left as “input” and the one in the right as “output”.
In the associated pattern, the projective operator P |+✓ i is applied to the qubits
on the left, but no operator is applied to the qubit on the right;
• (New Step 3 subcase) A vertex labeled “post-s” is contracted with any other
vertex label as one vertex with “post-s” label;
As a result, we obtain a projective pattern and associated open graph.
2. Apply local complementation [107] to turn postselected X measurements into postselected Z measurements. Remove from the graph the vertices being acted upon by a
postselected Z measurements (see Sec. 2.5.2).
3. Verify if the open graph obtained in Step 2 satisfies the generalized flow condition.
If it does, we obtain a runnable measurement pattern.
4. Translate the measurement pattern obtained in Step 3 to the circuit model using the
extended translation technique (Def. 14).
Note that if the open graph obtained in Step 2 satisfies the regular flow or the signalshifted flow conditions, more economical time-respecting circuits can be obtained by applying the compactification procedure in Algorithms 1 or 4, respectively, to the corresponding extended circuit.
As an example, I work out explicitly the stabilizer manipulations required to turn a
(probabilistic) BSS CTC simulation circuit into a deterministic one. This will be done
for the BSS circuit in Fig. 39.
144
i/o
h+|
|+i
|+i
i/o
i/o
i/o
J
| i
Mi✓ Zisi
i/o
h+|
i/o
|+i
P |+i
post-s
i
P |+i
o
i/o
P |+i
P |+
i
post-s
P |+i
i/o
(b)
(a)
(c)
Figure 41: Translating the BSS circuit in Fig. 39 to the one-way model (see Sec. 7.4.1).
7.4.1
Example
Let us now translate the circuit in Fig. 39 into a one-way measurement pattern using
the techniques described in Def. 23. In Fig. 41 we show the application of Step 1 in
Def. 23. As a result, we obtain a graph in Fig. 41-c and the associated projection-based
pattern:
|+i
|+i
|✓i
P1 P5 P3 |Gi,
|Gi ⌘ CZ34 CZ35 CZ32 CZ51 CZ12
N1 N2 N4 N5 |
in i3 ,
(7.11)
(7.12)
where the qubits’ sub-indices match those of the input qubits in the circuit of Fig. 39,
except qubit 5 which is a new qubit added in the pattern so as to implement the J
gate. Qubit 3 is our input state, and all the others are prepared in state |+ij by the Nj
command. The graph corresponding to state |Gi in Eq. (7.12) is shown in Fig. 42-a.
The projective X operators in Eq. (7.11) can be dealt with by using the so-called local
complementation rules (Step 2 in Def. 23) introduced in [107], which we now review.
Local complementation is an operation that changes a state in a way that is most
conveniently described by the change in its stabilizers. It corresponds to local unitaries
applied on a chosen qubit and its neighbors in the graph, and has been shown to preserve
the computation that can be performed using the state in the one-way model [60, 107, 97].
First, let us recall the definition of the phase gate S = |0ih0| + i|1ih1| and define the
p i⇡
i⇡
unitary C = HSH = 1/ 2(e 4 1 + e 4 X), where H is the one-qubit Hadamard gate. In
145
1 〈
1
2
2
+|
〈+|
3
5
4
C〈+| =
〈+|
S
4
(b)
C〈 i+| =〈 0|
2
2
S
5
3
S +| =〈 i+|
〈
SS
S
〈 i+|
3
5
4
4
(c)
SS
5
3
(a)
1
S +| =〈i+|
〈
(d)
2
C〈 i+| =〈 0|
3
2
3
5
4
4
(e)
(f)
Figure 42: Stabilizer manipulations that simplify the one-way sequence (7.11); hi +| denotes a projection onto |0i + i|1i. a) Graph representing initial entanglement structure
and operations. Qubit 3 is the input and qubits 1 and 5 are measured in the X basis. b)
Effect of local complementation (LC) on qubit 5; c) LC on qubit 1; d) Pauli Z deletion
of qubit 1; 3) LC on qubit 5; f) Pauli Z deletion of qubit 5. The final pattern represents
the one-way two-qubit implementation of the J( ✓) gate, see Fig. 37.
146
the manipulations that follow we will use the following identities:
SXS † = Y
(7.13)
CY C † = Z
(7.14)
CXC † = X
(7.15)
Let us now consider two different graph states | i and | i, specified below by listing their
stabilizers, respectively:
{f1 = X1 Z2 Z3 , f2 = Z1 X2 Z3 , f3 = Z1 Z2 X3 },
{f10 = X1 Z2 Z3 , f20 = Z1 X2 , f30 = Z1 X3 }.
(7.16)
(7.17)
It is easy to check that the unitary C1 S2† S3† applied on state | i changes the stabilizers as
follows:
{g1 = X1 Z2 Z3 , g2 = Y1 Y2 , g3 = Y1 Y3 }.
(7.18)
Note that the new stabilizers satisfy the relations f1 = g1 , f2 = g1 g2 and f3 = g1 g3 ,
hence C1 S2† S3† | i = | i. The application of local one-qubit unitaries such as C and
S does not change a state’s entanglement structure, but reveals that different one-way
patterns can correspond to the same entanglement resource. In our example, we know
states | i and | i satisfy the eigenvalue equations fi | i = | i and gi | i = | i. It is
straighforward to verify that these states can be represented by the two sequences of
commands | i = CZ12 CZ13 CZ23 N1 N2 N3 and | i = CZ12 CZ23 N1 N2 N3 , where Ni is the
preparation of qubit i in state |+i. Now we are able to illustrate graphically the local
complementation rule we described above by constructing the graphs associated with the
states | i and | i and relating them by the equality C1 S2† S3† | i = | i, as shown in Fig.
43.
The local complementation operation on non-input qubit i corresponds to applying
Q
the unitary Ci j2N (i) Sj† , with N (i) being the set of vertices (qubits) which are neigh-
bors of i in the entanglement graph. In addition to local complementation, we can also
delete a vertex from a graph by measuring it in the Z basis [97]. Z-deletion and local
complementation together change a state without altering the computation being performed [65, 97, 60, 107]. In general, we will need to apply these operations in graphs with
an arbitrary number of vertices and edges, choosing where the local complementation is
needed and applying the rule accordingly.
In Fig. 42 we illustrate a sequence of local complementations and Z-deletions that
transforms the initial sequence (7.11) (corresponding to the circuit of Fig. 39) into a
147
Figure 43: An analysis of the stabilizers in eqs. (7.16) and (7.17) indicates that the graph
on the right represents state | i and the one on the left state | i (see main text). The
local complementation unitary C1 S2† S3† changes the graph on the left into the graph on
the right. Local complementations change the state without changing its entanglement
structure or the one-way computations implementable by it.
simpler sequence implementing the same unitary. Starting from Fig. 42-a, we apply a
sequence of local complementations operations in order to remove the postselected Xmeasurements originated from the BSS protocol. The required operations are illustrated
in Fig. 42, so that in the end the postselected X-measurements become postselected Zmeasurements, which can be removed. This procedure results in the following, equivalent
command sequence:
|✓i
P3 CZ34 N2 N4 |
where |G0 i = CZ34 N4 |
in i3 .
in i3
|✓i
= P3 N2 |G0 i = M3✓ Z3s3 N2 |G0 i
(7.19)
Qubit 2 does not participate in the computation, as it
remains disentangled from the others. Therefore, the final projection-based pattern is
|✓i
(written using Pi
= Mi✓ Zisi ):
M3✓ Z3s3 |G0 i.
(7.20)
We see that elimination of the postselected X measurement results in exactly the same
command sequence of Eq. (7.2). As we have already seen, graph G0 has regular flow and
the associated measurement pattern is X4s3 M3✓ |G0 i, which implements deterministically
the unitary map J( ✓). To conclude the protocol given in Def. 23 we can translate the
computation back to the circuit model using the extended translation, which gives the
circuit in Fig. 12-a. The compact circuit implementing the same unitary can be obtained
by applying the compactification algorithm for regflow patterns (Alg. 1), which gives as
result the circuit in Fig. 12-b. With this, we have obtained a time-respecting, perfectly
physically implementable circuit from a CTC-assisted one.
148
ρ
ρ
CTC
CTC
ρ
U
in
ρ
in
V
ρout
(a)
U
ρout
(b)
Figure 44: a) Deutsch’s model for a CTC. b) This is the relationship between Deutsch’s
unitary U and unitary V in the BSS CTC circuit of Fig. 34-b.
7.5
Comparing the BSS model with Deutsch’s model
In this section I review the CTC model proposed by David Deutsch [39] and compare
the results obtained by applying BSS model to the circuit in Fig. 39 with the predictions
provided by Deutsch’s model when analyzing the same circuit. By doing so, I show that
Deutsch’s predictions are inconsistent with BSS’s.
7.5.1
The Deutsch model
In 1991 Deutsch [39] proposed a different, highly influential model for CTCs in quantum theory. Deutsch’s model avoids paradoxes by demanding self-consistent solutions
for the time-travelling systems. Let us start by analyzing a simple CTC-assisted circuit,
composed by just one time-respecting qubit and one time-travelling qubit, illustrated in
Fig. 44-a, where U is a general two-qubit unitary. The correspondence with BSS model
is shown in Fig. 44-b, so that Deutsch’s U = V · SW AP , with V being the unitary in
BSS’s formulation of CTCs (Fig. 34).
No paradox arises if we demand that the time-travelling qubit state ⇢CT C be a fixed
point of the dynamics:
⇥
⇤
⇢CT C = T rT R U (⇢CT C ⌦ ⇢in )U † ,
(7.21)
where the partial trace is over the time-respecting qubit. This self-consistency requirement
defines multiple solutions for ⇢CT C , each of which corresponds to a (generally non-linear)
map on ⇢in , which can be worked out from the solution ⇢CT C :
⇥
⇤
⇢out = T rCT C U (⇢CT C ⌦ ⇢in )U † .
where the partial trace is over the time-traveling qubit.
(7.22)
149
J
J
U
(a)
V
(b)
J
(c)
Figure 45: Three representations for the same CTC circuit. a) Deutsch formulation. b)
BSS formulation. c) Short-hand form of either.
There exists at least one fixed point to Eq. (7.21) and therefore there is always a ⇢CT C
satisfying the dynamics. This is so because every completely-positive map of the form
L[⇢] = T rA [U (⇢ ⌦ ⇢A )U † ] always has at least one solution. For scenarios where the selfconsistency condition in Eq. (7.21) has more than one solution, Deutsch has postulated a
“maximum entropy rule” stating that the solution with maximum entropy should be the
one chosen.
Moreover, note that although the self-consistency equations preserve the state of the
time-traveling qubit, the same doesn’t happen to its correlations: the qubit that emerges
into the past (after traveling through a CTC) does not preserve the (possibly) existing
correlations with qubits elsewhere in the Universe. It is conceptually different from what
happens in the BSS model, where both state and correlations of the time-traveling qubit
are preserved. As a consequence, computers interacting with Deutschian CTCs are considerably more powerful than computers interacting with BSS CTCs. A quantum computer
with access to Deutschian CTCs would be able to solve PSPACE-complete problems [7], a
complexity class that strictly contains PP-complete problems - the type of problems that
a quantum computer interacting with BSS CTCs is able to solve.
7.5.2
Comparison with the BSS model
Let us now study the same CTC we analyzed using BSS model, but now using
Deutsch’s by setting U = [J( ✓) ⌦ 1] · CZ · SW AP in Fig. 44. In the BSS model this corresponded to the circuit in Fig. 36. Three graphical representations for the same CTC are
shown in Fig. 45. We can represent ⇢in (~n) = 1/2(1 + ~n · ~ ) and ⇢CT C (m)
~ = 1/2(1 + m
~ · ~ ),
using the Pauli matrices ~ = (X, Y, Z). A simple calculation using Eqs. (7.21) gives us
the consistency conditions:
150
ρCTC
J-Θ
in
ρanc
|+〉
ρout
ρ
U
Figure 46: Deutsch’s formulation of the extended CTC circuit of Fig. 38-b.
(7.23)
mx = nz ,
ny cos ✓),
(7.24)
mz = mz (nx cos ✓ + ny sin ✓).
(7.25)
my = mz (nx sin ✓
The output state is ⇢out (~r) = 1/2(1 + ~r · ~ ), with ~r = (mx nz , my nz , mz ) being a function
of self-consistently assigned mx , my , mz .
There are two classes of self-consistent solutions to Eqs. (7.23)-(7.25). The first is
obtained by setting mz = 0, which yields a unique self-consistent ⇢CT C for each input
state:
⇢CT C : m
~ = (nz , 0, 0),
(7.26)
⇢out : ~r = (n2z , 0, 0).
(7.27)
These solutions are valid for all input states ⇢in (~n). The second class of solutions is
obtained by assuming that mz 6= 0 in Eqs. (7.23)-(7.25). Self-consistency dictates that
such solutions exist only for the particular ⇢in = |+✓ ih+✓ |, with ⇢CT C = ⇢out described by
m
~ = (0, 0, mz ).
Our analysis of the BSS model for this CTC considered not only the circuit where
the CTC appears on its own (Fig. 45), but also an enlarged circuit where the CTC acts
on only part of a larger entangled state (Fig. 38). For a fair comparison between the two
models, this can be done also in the Deutsch model, see Fig. 46. In this second approach
Deutsch’s unitary U is a three-qubit unitary encompassing all gates in the circuit of Fig.
38-b. A straightforward calculation using the self-consistency conditions in Eqs. (7.21)
and (7.22) yields the solution in which ⇢CT C and ⇢out are parametrized by m
~ = (nz , 0, 0),
with the ancilla qubit in output state ⇢anc (~a) : ~a = (n2z , 0, 0).
151
Our results for both the smaller circuit of Fig. 45-a and the larger circuit in Fig. 46
show that Deutsch’s model fails to implement the same input-output map as BSS and
the one-way model. The map implemented is the same only for the particular input state
⇢in = |0ih0|, with ⇢CT C = ⇢out = |+ih+|. For this input state Deutsch’s CTC qubit is not
entangled with the time-respecting qubits.
This suggests that the root of the problem with Deutsch’s model is the incompleteness
of the description of the CTC qubit. Deutsch’s model prescribes that the CTC qubit be
sent back in time as a mixed density matrix, which results in information loss about
its prior interactions. This is naturally taken care of in BSS model, as teleportation
preserves the CTC qubit’s entanglement with other systems, which was created via the
unitary interaction V .
In the recent papers [92, 93] Ralph and Myers have proposed an extension of Deutsch’s
model, in which one can heuristically describe the situation from the point of view of the
time-travelling qubit, which interacts with an infinite number of copies of itself. By adding
decoherence to this system, a unique solution is selected out of Deutsch’s possibly many
self-consistent solutions. This extension of Deutsch’s model is not, however, sufficient
to make it compatible with the predictions of the BSS model and one-way quantum
computation. A simple way to see that is to note that according to BSS the CTC output
is always in a pure state, whereas the multiple solutions proposed by Deutsch are typically
mixed.
7.5.3
Discussion
I have worked out the predictions of Deutsch’s and BSS models for the same CTC
examples and found a general disagreement between Deutsch’s predictions and what is
expected from the BSS/one-way model. This incompatibility stems from Deutsch’s incomplete description of the state being sent back in time, whose complete history of previous
interactions is preserved by the teleportation step used in BSS model.
One of the main conceptual differences between Deutsch’s and BSS models are the
self-consistency conditions. Deutsch’s model imposes that the density matrix description
of the qubit entering the CTC should be the same as the one that exits it (in the past).
Therefore, a measurement performed in the time-traveling qubit in the future (entering
the CTC) yields the same measurement statistics as if the measurement is performed in
the same qubit in the past (that is, entering the CTC). On the other hand, the BSS model
demands not only that such measurements yield the same statistics for the time-traveling
152
qubit but also that it should preserve its correlation with any other time-respecting qubits.
In other words, if a qubit interacts with another qubit and becomes correlated with it
before entering a CTC, the time-traveling qubit will exhibit the same correlations with
the time-respecting qubit when it exits the CTC in the past.
The different self-consistency requirements in those models also yield different computation capabilities. The BSS model, which simulates the action of CTCs by performing quantum measurements and then post-selecting the results, can solve problems in
the complexity class called post-BQP, or postselected quantum polynomial time. By the
post-BQP = PP theorem [5], we say that BSS CTCs can solve PP-complete problems, which
are a considerably powerful complexity class that strictly contains, for instance, the NP
class. On the other hand, Deutsch’s self consistency conditions requires Nature to solve
a fixed point problem, which is known to be a PSPACE-complete problem. Therefore,
Deutschian CTCs would in principle allow us to solve any problem in PSPACE, which
properly contains the complexity class PP (class of problems that a BSS CTC would be
able to solve).
In [75], Lloyd et al. worked out a link between CTCs as modeled by the BSS model
and path-integral formulations of quantum mechanics. Indeed, path-integral formulations
are specially suited for describing quantum field theories in curved spaces, a fact that
resonates with the idea that the BSS model is more adequate for describing the power
and action of CTCs. In particular, Lloyd et al. show in [75] that the dynamics provided by
the BSS model coincides with the path-integral description of fermions using Grassmann
fields given by Politzer [91]. In contrast, Deutsch’s model predictions are incompatible
with Politzer’s path-integral approach [91].
7.6
One-way model: a toy model for space-time?
I would like to conclude this chapter with an interesting possible application of the
1WQC model, namely its use as a toy model for spacetime. One of the main problems in
modern physics is the attempt to unify quantum mechanics, which describes three of the
four known fundamental interactions (electromagnetism, weak nuclear force and strong
nuclear force), with general relativity, which describes the forth fundamental interaction,
gravity. Some of the approaches to unify those theories suggest that spacetime would not
be independently constructed but rather a consequence of the laws of quantum mechanics.
And it is in the line of those approaches that the one-way model for quantum computing
153
might have some role to play.
In [98, 99], R. Raussendorf et al. argue that the one-way quantum computation model
has some of the properties which are expected in toy models able to generate spacetime
structures from none. Firstly, time in 1WQC appears as binary relations (which induces
the partial order) between quantum measurements (spacetime events), defined according
to the geometry of the associated graph. Therefore, time is not an external parameter
in the one-way model. Secondly, the correcting strategies defined by the flow theorems
(Sec. 3.3) have as their main purpose guaranteeing that the logical processing of the
computation is not affected by the inherent randomness of quantum measurements. In
other words, a quantum mechanical principle, namely the random character of quantum
measurements, is what induces the time-ordering in the one-way model.
Consider the following question: given a resource state |Gi (a graph state), how much
of temporal order can be defined directly from its geometry? If |Gi is obtained from a
quantum circuit (using, for instance, the methods described in Chapter 4), the temporal
order can be easily inferred by comparison with the gate ordering in the associated quantum circuit. Let us suppose, however, that |Gi is given with no reference to any kind of
computation. As we have seen many times in this thesis, if the input and output sets are
specified then the temporal order can be fully determined (by a flow theorem; See Sec.
3.3), even though we have not yet decided which computation is going to be performed
using the resource state. However, not every (I, O) pair induces a possible partial order.
In [80], the authors provide a way to determine which (I, O) pairs lead to a well-defined
temporal structure. In other words, time in a 1WQC (partial order between measurement
events) is almost entirely imposed by the geometry of the underlying graph; the missing
element being just a specification of the set of first qubits to be measured (inputs) and
the set of the last qubits to be measured (outputs).
One of the main results in [99] is a 1WQC analogue of Malament’s theorem [77],
which states that in any spacetime manifold the light cones determine the metric up to
a conformal factor. In the 1WQC model, the computation is usualy specified by three
elements: (i) resource state, (ii) measurement angles and (iii) classical processing relations,
which give the partial order between measurement events. This specification can be
further reduced since Theorem 4 in [99] states that (iii) completely determines (i). That is,
the classical processing relations uniquely defines the resource state. Hence, the classical
processing relations uniquely determines a one-way computation, up to the measurement
angles. This is analogous to the Malament’s theorem if the following identifications can
154
be inferred: (a) The light cones in general relativity correspond to classical processing
relations (that is, the measurement outcomes that a qubit i depends on is the backward
cone of i and the set of measurements that depend on qubit i’s measurement outcome
is the forward cone of i) and (b) the conformal factors in the spacetime metric at every
spacetime point are the angles of the measurements performed at every measurement
location (graph vertices).
In summary, the one-way model is able to mimic some of the properties of general
relativity. A 1WQC analog of Malament’s theorem was given in [99], where the authors
also discuss backward and forward light cones (mainly based on flow-like set of conditions)
and horizon of events. Those identifications between 1WQC and general relativity were
based in the principle that the randomness of measurement outcomes should not affect
the computation being performed, that is, the logical processing. It remains as an open
question, however, to determine what could be the analogue of this principle in more
general contexts, other than quantum computing. My work on closed timelike curves in
the one-way model [41] (which predated [99]), explored in this chapter, goes in the same
direction as [99], suggesting that 1WQC is indeed a good candidate as toy model for (at
least some) general relativity phenomena.
155
8
Conclusion and further research
directions
In this thesis I presented several results regarding the translation between the one-way
quantum computation model and the circuit, gate-array model. By proposing a different
approach for the translation of measurement patterns to the circuit model, I developed
a complete optimization procedure that works for arbitrary quantum circuits. Moreover,
using 1WQC techniques I have contributed to the discussion about the power of CTCassisted quantum circuits, suggesting that the little known (at the time of [41]) CTC
model by Bennett, Schumacher and Svetlichny gives more consistent predictions than the
one proposed by Deutsch in his highly influential paper [39].
One of the goals I tried to achieve in this thesis is to give a more complete introduction
to the so-called flow theorems. Those theorems characterize which graph states are needed
to implement a given deterministic computation and give a prescription for doing it. It
is clear that those theorems - and, more generally, the whole concept of deterministic
computation in measurement-based models - constitute an important piece of knowledge
for everyone interested in implementing 1WQC algorithms in physical systems. However,
the flow theorems are little known in the physics community, mainly because most of the
papers on the subject are published in computer science journals (and therefore written
for a computer science audience). I hope I have contributed to make this subject more
accessible and interesting to the physics community in Chapters 2 to 4, where I give an
introduction to the 1WQC model with special attention to the determinism theorems.
In a series of results the key concepts of regflow, gflow, maximally delayed gflow,
focused gflow and information preserving flow were introduced [33, 35, 24, 81, 80]. They
address the general question of determinism in MBQC while exploring different forms
of optimizations allowed by the model. I continued this line of research by introducing
a new type of flow - the signal-shifted flow [42] - and exploring several of its structural
properties. One of the most interesting properties of SSF is its parallel structure, which
156
equals in terms of depth the maximally delayed generalized flow [81]), which is the optimal
flow. It is worth noting that the link between signal-shifted flow and maximally delayed
gflow leads to a new efficient procedure for finding the optimal gflow of graphs with
regflow, since the algorithm for finding the SSF of a graph with regflow is more efficient
than the one for finding the maximally delayed gflow.
In Chapter 5, I propose a new way of translating 1WQC algorithms into quantum
circuits. In this new framework I first obtain the extended translation of the measurement
pattern one wants to translate, and then apply the so-called compactification procedure.
A compactification procedure explores the correcting structure of the translated measurement pattern in order to rewrite the extended circuit, allowing the removal of several
wires. Therefore, since the correcting structure of a measurement pattern is associated to
the type of flow the associated graph satisfies, a compactification procedure must be derived for each type of flow. In Sec. 5.2, I analyzed the structural properties of the regflow
function and designed an algorithm able to implement a compactification procedure for
regflow extended circuits (Algorithm 1). Then, in Sec. 5.3, I did the same for the recently
introduced signal-shifted flow, resulting in a compactification procedure for SSF extended
circuits (Algorithm 4).
I have also explored a few examples of more general measurement patterns, namely
the patterns associated with gflows that are different from regflow and SSF. Two complete examples are worked out in Sec. 5.4, where I also discuss the difficulty in designing a
compactification procedure for arbitrary gflows; the existence of such a general compactification procedure remains as an interesting open question. Moreover, my scheme fails to
compactify the parallel pattern obtained via Pauli simplification rules [23]. In other words
one needs to keep the extra space in order to preserve the parallel depth obtained due to
the Pauli measurements. This further indicates the crucial role that Clifford computation
plays in obtaining the superior parallel power of 1WQC over quantum circuits.
The SSF compactification algorithm was designed in such a way that it preserves the
reduced number of time slices J of SSF extended circuits. This allow us to use it for
optimizing quantum circuits in the following way. First, we translate a circuit to the
one-way model (as a regflow pattern) and then optimize it using the signal-shifting rules.
The optimized pattern is then translated to the circuit model as a SSF extended circuit,
from which one can obtain a compact circuit using our SSF compactification algorithm
(Algorithm 4). This optimization procedure, called Signal Shifting Circuit Optimization
(SSCO), is introduced in Chapter 6, where a complete example is worked out. Therein, I
157
use other translation and optimization techniques to compose an automated optimization
procedure that explores the circuit’s global structure to parallelize several J-gates and to
group together several CX gates, which allow further optimizations to take place using
specific method for Clifford gate parallelization [84].
Finally, in Chapter 7, I have shown how CTCs appear in a natural way in the one-way
model of measurement-based quantum computation. I studied a simple example of such
CTCs using the Bennett/Schumacher/Svetlichny (BSS) CTC model, whose predictions
agreed with those required by the one-way model. Going beyond the simple example we
studied, I have characterized a class of CTC circuits that admit deterministic BSS model
simulations. The simulations can be found using stabilizer techniques associated with
the one-way model, i.e., using the flow theorems. I have also worked out the predictions
of Deutsch’s model for the same CTC example and found a general disagreement in
comparison with what is expected from the BSS/one-way model. This incompatibility
stems from Deutsch’s incomplete description of the state being sent back in time, whose
complete history of previous interactions is preserved by the teleportation step used in
BSS model.
In the future, it would be interesting to characterize directly in the circuit model the
class of CTC-assisted circuits for which an efficient description using only time-respecting
qubits is possible. That could be found by re-interpreting in terms of circuit elements
the (necessary and sufficient) conditions for determinism required by the generalized flow.
Thus, instead of translating CTC-assisted circuits to the 1WQC model and then verifying
if the associated graphs satisfy the gflow conditions, one could determine directly in the
circuit model whether a CTC-assisted circuit could be deterministically simulated.
It would be interesting to apply the automated circuit optimization procedure developed in Chapter 6 to known quantum algorithms, comparing the gains with other known
optimization methods unrelated to MBQC.
In more general terms, an interesting problem that could benefit from our translation
approach is the study of restricted classes of computation. Recently, the study of classes
of computations that lie between the classical and quantum - that is, non-universal classes
of quantum computations that cannot be simulated efficiently in a classical computer has become an important line of investigation; examples of those classes are the class
of instantaneous quantum computation (IQP) [115] and those associated to the Boson
Sampling problem [116, 117, 118, 119, 120]. It is an interesting exercise to see how
the different restrictions translate between models, which may bring new insights on the
158
problem of quantum simulability.
An example where the translation to the 1WQC model has led to interesting results
is the class of Clifford computations, which can be implemented in a single computational time slice in the one-way model. On the other hand, it is known that there are
examples of single-round computations in the one-way model which do not translate as
Clifford circuits. Hopefully, the translation methods developed in this thesis can aid in
understanding better these new classes of interesting computations with finite depth.
Finally, as I have already mentioned, the development of a compactification procedure
able to preserve (at least partially) the optimization provided by more general correcting
strategies - in special those associated to arbitrary generalized flows - remains as an open
question. It is clear that such a procedure would help in attacking the problems that I
have raised here.
159
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